garid - brismu bridi

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Theorem garid 168

Description: Nested deduction form of gar 163 (Contributed by la korvo, 14-Jul-2025.)

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

**Assertion**
Ref	Expression
garid	⊢ ganai broda gi ganai ga brode gi brodo gi brodi

**Proof of Theorem garid**
Step	Hyp	Ref	Expression
1		garid.0	. . . 4 ⊢ ganai broda gi ganai brode gi brodi
2	1	ganai-swap12 29	. . 3 ⊢ ganai brode gi ganai broda gi brodi
3		garid.1	. . . 4 ⊢ ganai broda gi ganai brodo gi brodi
4	3	ganai-swap12 29	. . 3 ⊢ ganai brodo gi ganai broda gi brodi
5	2, 4	ga-sum 167	. 2 ⊢ ganai ga brode gi brodo gi ganai broda gi brodi
6	5	ganai-swap12 29	1 ⊢ ganai broda gi ganai ga brode gi brodo gi brodi

Colors of variables: sumti selbri bridi

Syntax hints: ganai bgan 9 ga bga 160

This theorem was proved from axioms: ax-mp 10 ax-k 11 ax-s 15 ax-ge-le 48 ax-ge-re 49 ax-ge-in 50

This theorem depends on definitions: df-go 83 df-ga 161

This theorem is referenced by: garidan 169