Theorem eq2tri 2256

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

Hypotheses

Ref	Expression
eq2tr.1	⊢ (𝐴 = 𝐶 → 𝐷 = 𝐹)
eq2tr.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 266	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (𝐵 = 𝐷 ∧ 𝐴 = 𝐶))
2		eq2tr.1	. . . 4 ⊢ (𝐴 = 𝐶 → 𝐷 = 𝐹)
3	2	eqeq2d 2208	. . 3 ⊢ (𝐴 = 𝐶 → (𝐵 = 𝐷 ↔ 𝐵 = 𝐹))
4	3	pm5.32i 454	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐹))
5		eq2tr.2	. . . 4 ⊢ (𝐵 = 𝐷 → 𝐶 = 𝐺)
6	5	eqeq2d 2208	. . 3 ⊢ (𝐵 = 𝐷 → (𝐴 = 𝐶 ↔ 𝐴 = 𝐺))
7	6	pm5.32i 454	. 2 ⊢ ((𝐵 = 𝐷 ∧ 𝐴 = 𝐶) ↔ (𝐵 = 𝐷 ∧ 𝐴 = 𝐺))
8	1, 4, 7	3bitr3i 210	1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐹) ↔ (𝐵 = 𝐷 ∧ 𝐴 = 𝐺))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189

This theorem is referenced by:  xpassen  6889