Theorem syl5req 2398

Description: An equality transitivity deduction. (Contributed by NM, 29-Mar-1998.)

Hypotheses

Ref	Expression
syl5req.1	⊢ A = B
syl5req.2	⊢ (φ → B = C)

Assertion

Ref	Expression
syl5req	⊢ (φ → C = A)

Proof of Theorem syl5req

Step	Hyp	Ref	Expression
1		syl5req.1	. . 3 ⊢ A = B
2		syl5req.2	. . 3 ⊢ (φ → B = C)
3	1, 2	syl5eq 2397	. 2 ⊢ (φ → A = C)
4	3	eqcomd 2358	1 ⊢ (φ → C = A)

Syntax hints:   → wi 4   = wceq 1642

This theorem was proved from 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 theorem depends on definitions:  df-bi 177  df-ex 1542  df-cleq 2346

This theorem is referenced by:  syl5reqr  2400  nnsucelrlem3  4427  tfinprop  4490  funcnvres  5166