Theorem eq2tri 2800

Description: A compound transitive inference for class equality. (Contributed by NM, 22-Jan-2004.)

Hypotheses

Ref	Expression
eq2tri.1	⊢ (𝐴 = 𝐶 → 𝐷 = 𝐹)
eq2tri.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 462	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (𝐵 = 𝐷 ∧ 𝐴 = 𝐶))
2		eq2tri.1	. . . 4 ⊢ (𝐴 = 𝐶 → 𝐷 = 𝐹)
3	2	eqeq2d 2744	. . 3 ⊢ (𝐴 = 𝐶 → (𝐵 = 𝐷 ↔ 𝐵 = 𝐹))
4	3	pm5.32i 576	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐹))
5		eq2tri.2	. . . 4 ⊢ (𝐵 = 𝐷 → 𝐶 = 𝐺)
6	5	eqeq2d 2744	. . 3 ⊢ (𝐵 = 𝐷 → (𝐴 = 𝐶 ↔ 𝐴 = 𝐺))
7	6	pm5.32i 576	. 2 ⊢ ((𝐵 = 𝐷 ∧ 𝐴 = 𝐶) ↔ (𝐵 = 𝐷 ∧ 𝐴 = 𝐺))
8	1, 4, 7	3bitr3i 301	1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐹) ↔ (𝐵 = 𝐷 ∧ 𝐴 = 𝐺))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-cleq 2725

This theorem is referenced by:  xpassen  9063