Theorem sylan9req 2292

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 2244	. 2 |- ( ph -> A = B )
3		sylan9req.2	. 2 |- ( ps -> B = C )
4	2, 3	sylan9eq 2291	1 |- ( ( ph /\ ps ) -> A = C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-cleq 2231

This theorem is referenced by:  fndmu  5479  fodmrnu  5618  funcoeqres  5665  fvunsng  5900  mapsnd  6960  prarloclem5  7857  addlocprlemeq  7890  zdiv  9713  resqrexlemnm  11762  fprodssdc  12335  dvdsmulc  12564  cncongrcoprm  12862  mgmidmo  13669  lgsmodeq  16078