Theorem eqeqan12rd 2187

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 2186	. 2 |- ( ( ph /\ ps ) -> ( A = C <-> B = D ) )
4	3	ancoms 266	1 |- ( ( ps /\ ph ) -> ( A = C <-> B = D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348

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 1440  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163

This theorem is referenced by:  omp1eomlem  7071