Theorem eqeqan12rd 1528

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 1527	. 2 |- ((ph /\ ps) -> (A = C <-> B = D))
4	3	ancoms 438	1 |- ((ps /\ ph) -> (A = C <-> B = D))

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221   = wceq 988

This theorem is referenced by:  fvopab4gf 3857  fvopabgf 3863  fvopabnf 3864  tfrlem5 3991  inf3lema 4695  numth 4870  zorn2 4882  fsumcnlem 8109  effoi 8864

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 995  ax-17 1003  ax-4 1005  ax-5o 1007  ax-ext 1494

This theorem depends on definitions:  df-bi 145  df-an 223  df-cleq 1505

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