Theorem syl5req 2085

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 2084	. 2 ⊢ (𝜑 → 𝐴 = 𝐶)
4	3	eqcomd 2045	1 ⊢ (𝜑 → 𝐶 = 𝐴)

Syntax hints:   → wi 4   = wceq 1243

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400  ax-17 1419  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033

This theorem is referenced by:  syl5reqr  2087  opeqsn  3989  relop  4486  funopg  4934  funcnvres  4972  apreap  7578  recextlem1  7632  nn0supp  8234  intqfrac2  9161