Theorem eq2tri 2822

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 466	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (𝐵 = 𝐷 ∧ 𝐴 = 𝐶))
2		eq2tri.1	. . . 4 ⊢ (𝐴 = 𝐶 → 𝐷 = 𝐹)
3	2	eqeq2d 2771	. . 3 ⊢ (𝐴 = 𝐶 → (𝐵 = 𝐷 ↔ 𝐵 = 𝐹))
4	3	pm5.32i 585	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐹))
5		eq2tri.2	. . . 4 ⊢ (𝐵 = 𝐷 → 𝐶 = 𝐺)
6	5	eqeq2d 2771	. . 3 ⊢ (𝐵 = 𝐷 → (𝐴 = 𝐶 ↔ 𝐴 = 𝐺))
7	6	pm5.32i 585	. 2 ⊢ ((𝐵 = 𝐷 ∧ 𝐴 = 𝐶) ↔ (𝐵 = 𝐷 ∧ 𝐴 = 𝐺))
8	1, 4, 7	3bitr3i 304	1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐹) ↔ (𝐵 = 𝐷 ∧ 𝐴 = 𝐺))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-cleq 2752

This theorem is used by:  xpassen  9094