Theorem eq2tr 1536

Description: A compound transitive inference for class equality.

Hypotheses

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

Assertion

Ref	Expression
eq2tr	|- ((A = C /\ B = F) <-> (B = D /\ A = G))

Proof of Theorem eq2tr

Step	Hyp	Ref	Expression
1		ancom 437	. 2 |- ((A = C /\ B = D) <-> (B = D /\ A = C))
2		eq2tr.1	. . . 4 |- (A = C -> D = F)
3	2	eqeq2d 1489	. . 3 |- (A = C -> (B = D <-> B = F))
4	3	pm5.32i 647	. 2 |- ((A = C /\ B = D) <-> (A = C /\ B = F))
5		eq2tr.2	. . . 4 |- (B = D -> C = G)
6	5	eqeq2d 1489	. . 3 |- (B = D -> (A = C <-> A = G))
7	6	pm5.32i 647	. 2 |- ((B = D /\ A = C) <-> (B = D /\ A = G))
8	1, 4, 7	3bitr3 181	1 |- ((A = C /\ B = F) <-> (B = D /\ A = G))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958

This theorem is referenced by:  xpcomen 4445  xpassen 4447

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-17 973  ax-4 975  ax-5o 977  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-cleq 1472

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