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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402

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

This theorem is used by:  fndmu  5487  fodmrnu  5626  funcoeqres  5673  fvunsng  5912  mapsnd  6974  prarloclem5  7873  addlocprlemeq  7906  zdiv  9745  resqrexlemnm  11806  fprodssdc  12382  dvdsmulc  12611  cncongrcoprm  12909  mgmidmo  13751  lgsmodeq  16369