Theorem eqeqan12rd 1538

Description: A useful inference for substituting definitions into an equality.

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 1537	. 2 |- ((ph /\ ps) -> (A = C <-> B = D))
4	3	ancoms 447	1 |- ((ps /\ ph) -> (A = C <-> B = D))

Syntax hints:   -> wi 3   <-> wb 153   /\ wa 230   = wceq 997

This theorem is referenced by:  fvopab4gf 3838  fvopabgf 3844  fvopabnf 3845  tfrlem5 3973  inf3lema 4671  numth 4846  zorn2 4858  fsumcnlem 8074  effoi 8828  eigorthi 9846

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1004  ax-17 1012  ax-4 1014  ax-5o 1016  ax-ext 1504

This theorem depends on definitions:  df-bi 154  df-an 232  df-cleq 1515

Copyright terms: Public domain