Theorem eq2tri 2412

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 437	. 2 ⊢ ((A = C ∧ B = D) ↔ (B = D ∧ A = C))
2		eq2tr.1	. . . 4 ⊢ (A = C → D = F)
3	2	eqeq2d 2364	. . 3 ⊢ (A = C → (B = D ↔ B = F))
4	3	pm5.32i 618	. 2 ⊢ ((A = C ∧ B = D) ↔ (A = C ∧ B = F))
5		eq2tr.2	. . . 4 ⊢ (B = D → C = G)
6	5	eqeq2d 2364	. . 3 ⊢ (B = D → (A = C ↔ A = G))
7	6	pm5.32i 618	. 2 ⊢ ((B = D ∧ A = C) ↔ (B = D ∧ A = G))
8	1, 4, 7	3bitr3i 266	1 ⊢ ((A = C ∧ B = F) ↔ (B = D ∧ A = G))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346

This theorem is used by: (None)