Theorem eq2tri 2412

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 437	. 2 |- ((A = C /\ B = D) <-> (B = D /\ A = C))
2		eq2tr.1	. . . 4 |- (A = C -> D = F)
3	2	eqeq2d 2364	. . 3 |- (A = C -> (B = D <-> B = F))
4	3	pm5.32i 618	. 2 |- ((A = C /\ B = D) <-> (A = C /\ B = F))
5		eq2tr.2	. . . 4 |- (B = D -> C = G)
6	5	eqeq2d 2364	. . 3 |- (B = D -> (A = C <-> A = G))
7	6	pm5.32i 618	. 2 |- ((B = D /\ A = C) <-> (B = D /\ A = G))
8	1, 4, 7	3bitr3i 266	1 |- ((A = C /\ B = F) <-> (B = D /\ A = G))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346

This theorem is referenced by: (None)