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Theorem go-ganai 54

Description: Biconditional implication may be weakened to unidirectional implication. Category-theoretically, this theorem embeds the core of Loj. Inference form of left side of goli 53. Theorem biimpi in [ILE] p. 0. (Contributed by la korvo, 17-Jul-2023.) (Shortened by la korvo, 29-Jul-2023.)

Hypothesis

Ref	Expression
go-ganai.0	⊢ go broda gi brode

Assertion

Ref	Expression
go-ganai	⊢ ganai broda gi brode

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Proof of Theorem go-ganai

Step	Hyp	Ref	Expression
1		go-ganai.0	. . 3 ⊢ go broda gi brode
2	1	goli 53	. 2 ⊢ ge ganai broda gi brode gi ganai brode gi broda
3	2	ge-lei 37	1 ⊢ ganai broda gi brode

Colors of variables: sumti selbri bridi

Syntax hints:  ganai bgan 9

This theorem was proved from axioms:  ax-mp 10  ax-ge-le 34

This theorem depends on definitions:  df-go 52

This theorem is referenced by:  bi  69  bi-rev-syl  71  sylbi  72  sylib  73  syl5bi  76  se-dual  172  se-dual-l  173  se-ganaii  175  se-ganair  176  ro2-bi  194  ro2-bi-rev  195  naku-uncur  212  te-dual  327  te-dual-l  328  te-ganaii  330  te-ganair  331  sub1  352  mapti-ckini  581

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