Theorem syl6req 2402

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

Hypotheses

Ref	Expression
syl6req.1	⊢ (φ → A = B)
syl6req.2	⊢ B = C

Assertion

Ref	Expression
syl6req	⊢ (φ → C = A)

Proof of Theorem syl6req

Step	Hyp	Ref	Expression
1		syl6req.1	. . 3 ⊢ (φ → A = B)
2		syl6req.2	. . 3 ⊢ B = C
3	1, 2	syl6eq 2401	. 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:  syl6reqr  2404  lefinlteq  4463  elxp4  5108  mapsn  6026