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Theorem eq2tri 2821
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 465 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) ↔ (𝐵 = 𝐷𝐴 = 𝐶))
2 eq2tr.1 . . . 4 (𝐴 = 𝐶𝐷 = 𝐹)
32eqeq2d 2770 . . 3 (𝐴 = 𝐶 → (𝐵 = 𝐷𝐵 = 𝐹))
43pm5.32i 672 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐹))
5 eq2tr.2 . . . 4 (𝐵 = 𝐷𝐶 = 𝐺)
65eqeq2d 2770 . . 3 (𝐵 = 𝐷 → (𝐴 = 𝐶𝐴 = 𝐺))
76pm5.32i 672 . 2 ((𝐵 = 𝐷𝐴 = 𝐶) ↔ (𝐵 = 𝐷𝐴 = 𝐺))
81, 4, 73bitr3i 290 1 ((𝐴 = 𝐶𝐵 = 𝐹) ↔ (𝐵 = 𝐷𝐴 = 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-cleq 2753
This theorem is referenced by:  xpassen  8219
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