Theorem sylan9req 2220

Description: An equality transitivity deduction. (Contributed by NM, 23-Jun-2007.)

Hypotheses

Ref	Expression
sylan9req.1	|- ( ph -> B = A )
sylan9req.2	|- ( ps -> B = C )

Assertion

Ref	Expression
sylan9req	|- ( ( ph /\ ps ) -> A = C )

Proof of Theorem sylan9req

Step	Hyp	Ref	Expression
1		sylan9req.1	. . 3 |- ( ph -> B = A )
2	1	eqcomd 2171	. 2 |- ( ph -> A = B )
3		sylan9req.2	. 2 |- ( ps -> B = C )
4	2, 3	sylan9eq 2219	1 |- ( ( ph /\ ps ) -> A = C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-4 1498  ax-17 1514  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158

This theorem is referenced by:  fndmu  5289  fodmrnu  5418  funcoeqres  5463  fvunsng  5679  prarloclem5  7441  addlocprlemeq  7474  zdiv  9279  resqrexlemnm  10960  fprodssdc  11531  dvdsmulc  11759  cncongrcoprm  12038  mgmidmo  12603  lgsmodeq  13586