Theorem syl5reqr 2128

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

Hypotheses

Ref	Expression
syl5reqr.1	⊢ 𝐵 = 𝐴
syl5reqr.2	⊢ (𝜑 → 𝐵 = 𝐶)

Assertion

Ref	Expression
syl5reqr	⊢ (𝜑 → 𝐶 = 𝐴)

Proof of Theorem syl5reqr

Step	Hyp	Ref	Expression
1		syl5reqr.1	. . 3 ⊢ 𝐵 = 𝐴
2	1	eqcomi 2085	. 2 ⊢ 𝐴 = 𝐵
3		syl5reqr.2	. 2 ⊢ (𝜑 → 𝐵 = 𝐶)
4	2, 3	syl5req 2126	1 ⊢ (𝜑 → 𝐶 = 𝐴)

Syntax hints:   → wi 4   = wceq 1284

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-4 1440  ax-17 1459  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074

This theorem is referenced by:  bm2.5ii  4240  resdmdfsn  4671  f1o00  5181  fmpt  5340  fmptsn  5373  resfunexg  5403  pm54.43  6459  prarloclem5  6690  recexprlem1ssl  6823  recexprlem1ssu  6824  iooval2  8938  resqrexlemover  9896