Theorem syl6req 2138

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

Hypotheses

Ref	Expression
syl6req.1	|- ( ph -> A = B )
syl6req.2	|- B = C

Assertion

Ref	Expression
syl6req	|- ( ph -> C = A )

Proof of Theorem syl6req

Step	Hyp	Ref	Expression
1		syl6req.1	. . 3 |- ( ph -> A = B )
2		syl6req.2	. . 3 |- B = C
3	1, 2	syl6eq 2137	. 2 |- ( ph -> A = C )
4	3	eqcomd 2094	1 |- ( ph -> 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  4931  elxp5  4932  fo1stresm  5946  fo2ndresm  5947  eloprabi  5980  fo2ndf  6006  xpsnen  6591  xpassen  6600  ac6sfi  6668  undifdc  6688  ine0  7926  nn0n0n1ge2  8871  fzval2  9481  fseq1p1m1  9562  fsum2dlemstep  10882  modfsummodlemstep  10905  ef4p  11038  sin01bnd  11102  odd2np1  11205  sqpweven  11485  2sqpwodd  11486