Theorem syl6req 2138

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 2137	. 2 ⊢ (𝜑 → 𝐴 = 𝐶)
4	3	eqcomd 2094	1 ⊢ (𝜑 → 𝐶 = 𝐴)

Syntax hints:   → wi 4   = wceq 1290

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-4 1446  ax-17 1465  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-cleq 2082

This theorem is referenced by:  syl6reqr  2140  elxp4  4933  elxp5  4934  fo1stresm  5948  fo2ndresm  5949  eloprabi  5982  fo2ndf  6008  xpsnen  6593  xpassen  6602  ac6sfi  6670  undifdc  6690  ine0  7935  nn0n0n1ge2  8880  fzval2  9490  fseq1p1m1  9571  fsum2dlemstep  10891  modfsummodlemstep  10914  ef4p  11047  sin01bnd  11111  odd2np1  11214  sqpweven  11494  2sqpwodd  11495