Theorem eq2tri 2230

Description: A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.)

Hypotheses

Ref	Expression
eq2tr.1	|- ( A = C -> D = F )
eq2tr.2	|- ( B = D -> C = G )

Assertion

Ref	Expression
eq2tri	|- ( ( A = C /\ B = F ) <-> ( B = D /\ A = G ) )

Proof of Theorem eq2tri

Step	Hyp	Ref	Expression
1		ancom 264	. 2 |- ( ( A = C /\ B = D ) <-> ( B = D /\ A = C ) )
2		eq2tr.1	. . . 4 |- ( A = C -> D = F )
3	2	eqeq2d 2182	. . 3 |- ( A = C -> ( B = D <-> B = F ) )
4	3	pm5.32i 451	. 2 |- ( ( A = C /\ B = D ) <-> ( A = C /\ B = F ) )
5		eq2tr.2	. . . 4 |- ( B = D -> C = G )
6	5	eqeq2d 2182	. . 3 |- ( B = D -> ( A = C <-> A = G ) )
7	6	pm5.32i 451	. 2 |- ( ( B = D /\ A = C ) <-> ( B = D /\ A = G ) )
8	1, 4, 7	3bitr3i 209	1 |- ( ( A = C /\ B = F ) <-> ( B = D /\ A = G ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163

This theorem is referenced by:  xpassen  6808