mpan9 - brismu bridi

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Theorem mpan9 76

Description: An inference. (Contributed by la korvo, 9-Jul-2025.)

**Hypotheses**
Ref	Expression
mpan9.0	⊢ ganai broda gi brode
mpan9.1	⊢ ganai brodi gi ganai brode gi brodo

**Assertion**
Ref	Expression
mpan9	⊢ ganai ge broda gi brodi gi brodo

**Proof of Theorem mpan9**
Step	Hyp	Ref	Expression
1		mpan9.0	. . 3 ⊢ ganai broda gi brode
2		mpan9.1	. . 3 ⊢ ganai brodi gi ganai brode gi brodo
3	1, 2	syl5 32	. 2 ⊢ ganai brodi gi ganai broda gi brodo
4	3	cur-swap12 60	1 ⊢ ganai ge broda gi brodi gi brodo

Colors of variables: sumti selbri bridi

This theorem was proved from axioms: ax-mp 10 ax-k 11 ax-s 15 ax-ge-le 48 ax-ge-re 49

This theorem is referenced by: sylan 77