Theorem eqeqan12rd 1491

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 1490	. 2 |- ((ph /\ ps) -> (A = C <-> B = D))
4	3	ancoms 436	1 |- ((ps /\ ph) -> (A = C <-> B = D))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956

This theorem is referenced by:  fvopab4gf 3781  fvopabgf 3787  fvopabnf 3788  tfrlem5 3915  inf3lema 4609  numth 4784  zorn2 4796  fsumcnlem 7989  effoi 8745  eigorth 9763

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-cleq 1469

Copyright terms: Public domain