ge-go - brismu bridi

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Theorem ge-go 111

Description: Conjunction implies biimplication. (Contributed by la korvo, 25-Jun-2024.)

**Hypothesis**
Ref	Expression
ge-go.0	⊢ ge broda gi brode

**Assertion**
Ref	Expression
ge-go	⊢ go broda gi brode

**Proof of Theorem ge-go**
Step	Hyp	Ref	Expression
1		ge-go.0	. . 3 ⊢ ge broda gi brode
2		ge-ganai 42	. . 3 ⊢ ganai ge broda gi brode gi ganai broda gi brode
3	1, 2	ax-mp 10	. 2 ⊢ ganai broda gi brode
4		ge-com-lem 109	. . . 4 ⊢ ganai ge broda gi brode gi ge brode gi broda
5	1, 4	ax-mp 10	. . 3 ⊢ ge brode gi broda
6		ge-ganai 42	. . 3 ⊢ ganai ge brode gi broda gi ganai brode gi broda
7	5, 6	ax-mp 10	. 2 ⊢ ganai brode gi broda
8	3, 7	iso 56	1 ⊢ go broda gi brode

Colors of variables: sumti selbri bridi

Syntax hints: ganai bgan 9 ge bge 33

This theorem was proved from axioms: ax-mp 10 ax-k 11 ax-s 14 ax-ge-le 34 ax-ge-re 35 ax-ge-in 36

This theorem depends on definitions: df-go 52

This theorem is referenced by: simsa-mintu 305