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Theorem eq2tri 2096
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 253 . 2 ((A = 𝐶 B = 𝐷) ↔ (B = 𝐷 A = 𝐶))
2 eq2tr.1 . . . 4 (A = 𝐶𝐷 = 𝐹)
32eqeq2d 2048 . . 3 (A = 𝐶 → (B = 𝐷B = 𝐹))
43pm5.32i 427 . 2 ((A = 𝐶 B = 𝐷) ↔ (A = 𝐶 B = 𝐹))
5 eq2tr.2 . . . 4 (B = 𝐷𝐶 = 𝐺)
65eqeq2d 2048 . . 3 (B = 𝐷 → (A = 𝐶A = 𝐺))
76pm5.32i 427 . 2 ((B = 𝐷 A = 𝐶) ↔ (B = 𝐷 A = 𝐺))
81, 4, 73bitr3i 199 1 ((A = 𝐶 B = 𝐹) ↔ (B = 𝐷 A = 𝐺))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-4 1397  ax-17 1416  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-cleq 2030
This theorem is referenced by:  xpassen  6240
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