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Theorem eq2tri 2174
 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 = 𝐶𝐵 = 𝐹) ↔ (𝐵 = 𝐷𝐴 = 𝐺))

Proof of Theorem eq2tri
StepHypRef Expression
1 ancom 264 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) ↔ (𝐵 = 𝐷𝐴 = 𝐶))
2 eq2tr.1 . . . 4 (𝐴 = 𝐶𝐷 = 𝐹)
32eqeq2d 2126 . . 3 (𝐴 = 𝐶 → (𝐵 = 𝐷𝐵 = 𝐹))
43pm5.32i 447 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐹))
5 eq2tr.2 . . . 4 (𝐵 = 𝐷𝐶 = 𝐺)
65eqeq2d 2126 . . 3 (𝐵 = 𝐷 → (𝐴 = 𝐶𝐴 = 𝐺))
76pm5.32i 447 . 2 ((𝐵 = 𝐷𝐴 = 𝐶) ↔ (𝐵 = 𝐷𝐴 = 𝐺))
81, 4, 73bitr3i 209 1 ((𝐴 = 𝐶𝐵 = 𝐹) ↔ (𝐵 = 𝐷𝐴 = 𝐺))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1314 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-4 1470  ax-17 1489  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-cleq 2108 This theorem is referenced by:  xpassen  6677
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