Theorem syl6req 2089

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 2088	. 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:  syl6reqr  2091  elxp4  4808  elxp5  4809  fo1stresm  5788  fo2ndresm  5789  eloprabi  5822  fo2ndf  5848  xpsnen  6295  xpassen  6304  ac6sfi  6352  ine0  7391  nn0n0n1ge2  8311  fzval2  8877  fseq1p1m1  8956