Theorem syl6reqr 2404

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

Hypotheses

Ref	Expression
syl6reqr.1	⊢ (φ → A = B)
syl6reqr.2	⊢ C = B

Assertion

Ref	Expression
syl6reqr	⊢ (φ → C = A)

Proof of Theorem syl6reqr

Step	Hyp	Ref	Expression
1		syl6reqr.1	. 2 ⊢ (φ → A = B)
2		syl6reqr.2	. . 3 ⊢ C = B
3	2	eqcomi 2357	. 2 ⊢ B = C
4	1, 3	syl6req 2402	1 ⊢ (φ → C = A)

This proof depends on syntax axioms:   → wi 4   = 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-ex 1542  df-cleq 2346

This theorem is used by:  iftrue  3669  iffalse  3670  difprsn1  3848  funimacnv  5169  dfimafn  5367  fniinfv  5373  fvco2  5383  fniunfv  5467  isoini  5498  dmmptg  5685  nchoicelem14  6303