Theorem sylan9req 2191

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 2143	. 2 |- ( ph -> A = B )
3		sylan9req.2	. 2 |- ( ps -> B = C )
4	2, 3	sylan9eq 2190	1 |- ( ( ph /\ ps ) -> A = C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331

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 1423  ax-gen 1425  ax-4 1487  ax-17 1506  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-cleq 2130

This theorem is referenced by:  fndmu  5219  fodmrnu  5348  funcoeqres  5391  fvunsng  5607  prarloclem5  7301  addlocprlemeq  7334  zdiv  9132  resqrexlemnm  10783  dvdsmulc  11510  cncongrcoprm  11776