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Theorem eq2tri 2412
 Description: A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.)
Hypotheses
Ref Expression
eq2tr.1 (A = CD = F)
eq2tr.2 (B = DC = G)
Assertion
Ref Expression
eq2tri ((A = C B = F) ↔ (B = D A = G))

Proof of Theorem eq2tri
StepHypRef Expression
1 ancom 437 . 2 ((A = C B = D) ↔ (B = D A = C))
2 eq2tr.1 . . . 4 (A = CD = F)
32eqeq2d 2364 . . 3 (A = C → (B = DB = F))
43pm5.32i 618 . 2 ((A = C B = D) ↔ (A = C B = F))
5 eq2tr.2 . . . 4 (B = DC = G)
65eqeq2d 2364 . . 3 (B = D → (A = CA = G))
76pm5.32i 618 . 2 ((B = D A = C) ↔ (B = D A = G))
81, 4, 73bitr3i 266 1 ((A = C B = F) ↔ (B = D A = G))
 Colors of variables: wff setvar class 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)
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