Theorem r19.15 1745

Description: Distribute a restricted universal quantifier over a biconditional. Theorem 19.15 of [Margaris] p. 90 with restricted quantification.

Assertion

Ref	Expression
r19.15	|- (A.x e. A (ph <-> ps) -> (A.x e. A ph <-> A.x e. A ps))

Proof of Theorem r19.15

Step	Hyp	Ref	Expression
1		hbra1 1679	. 2 |- (A.x e. A (ph <-> ps) -> A.xA.x e. A (ph <-> ps))
2		ra4 1686	. . 3 |- (A.x e. A (ph <-> ps) -> (x e. A -> (ph <-> ps)))
3	2	imp 350	. 2 |- ((A.x e. A (ph <-> ps) /\ x e. A) -> (ph <-> ps))
4	1, 3	ralbida 1649	1 |- (A.x e. A (ph <-> ps) -> (A.x e. A ph <-> A.x e. A ps))

Syntax hints:   -> wi 3   <-> wb 146   e. wcel 955  A.wral 1637

This theorem is referenced by:  rankonid 4667  kmlem8 4744  kmlem13 4749  expcnv 7168

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-4 970  ax-5o 972

This theorem depends on definitions:  df-bi 147  df-an 225  df-ral 1641

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