Theorem eqeqan12rd 2640

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 2638	. 2 |- ( ( ph /\ ps ) -> ( A = C <-> B = D ) )
4	3	ancoms 469	1 |- ( ( ps /\ ph ) -> ( A = C <-> B = D ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615

This theorem is referenced by:  fmptco  6396  axcontlem4  25847  usgredg4  26109  cusgrsize  26350  uspgr2wlkeqi  26544  clwwlksf1  26917  eigorthi  28696  expdiophlem2  37589  pwssplit4  37659  fmtnoodd  41445