Theorem eqeqan12rd 2255

Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.)

Hypotheses

Ref	Expression
eqeqan12rd.1	|- ( ph -> A = B )
eqeqan12rd.2	|- ( ps -> C = D )

Assertion

Ref	Expression
eqeqan12rd	|- ( ( ps /\ ph ) -> ( A = C <-> B = D ) )

Proof of Theorem eqeqan12rd

Step	Hyp	Ref	Expression
1		eqeqan12rd.1	. . 3 |- ( ph -> A = B )
2		eqeqan12rd.2	. . 3 |- ( ps -> C = D )
3	1, 2	eqeqan12d 2254	. 2 |- ( ( ph /\ ps ) -> ( A = C <-> B = D ) )
4	3	ancoms 268	1 |- ( ( ps /\ ph ) -> ( A = C <-> B = D ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = 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:  omp1eomlem  7434  usgredg4  16456  uspgr2wlkeqi  16608