Theorem cleq2tr 1148

Description: A compound transitive inference for class equality.

Hypotheses

Ref	Expression
cleq2tr.1	|- (A = C -> D = F)
cleq2tr.2	|- (B = D -> C = G)

Assertion

Ref	Expression
cleq2tr	|- ((A = C /\ B = F) <-> (B = D /\ A = G))

Proof of Theorem cleq2tr

Step	Hyp	Ref	Expression
1		ancom 333	. 2 |- ((A = C /\ B = D) <-> (B = D /\ A = C))
2		cleq2tr.1	. . . 4 |- (A = C -> D = F)
3	2	cleq2d 1112	. . 3 |- (A = C -> (B = D <-> B = F))
4	3	pm5.32i 489	. 2 |- ((A = C /\ B = D) <-> (A = C /\ B = F))
5		cleq2tr.2	. . . 4 |- (B = D -> C = G)
6	5	cleq2d 1112	. . 3 |- (B = D -> (A = C <-> A = G))
7	6	pm5.32i 489	. 2 |- ((B = D /\ A = C) <-> (B = D /\ A = G))
8	1, 4, 7	3bitr3 156	1 |- ((A = C /\ B = F) <-> (B = D /\ A = G))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091

This theorem is referenced by:  xpcomen 3343  xpassen 3344

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-cleq 1097

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