Theorem r19.15 1292

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 1237	. . 3 |- (A.x e. A (ph <-> ps) -> A.xA.x e. A (ph <-> ps))
2		id 9	. . 3 |- (A.x e. A (ph <-> ps) -> A.x e. A (ph <-> ps))
3	1, 2	19.21ai 740	. 2 |- (A.x e. A (ph <-> ps) -> A.xA.x e. A (ph <-> ps))
4		ra4 1243	. . 3 |- (A.x e. A (ph <-> ps) -> (x e. A -> (ph <-> ps)))
5	4	imp 277	. 2 |- ((A.x e. A (ph <-> ps) /\ x e. A) -> (ph <-> ps))
6	3, 5	biralda 1213	1 |- (A.x e. A (ph <-> ps) -> (A.x e. A ph <-> A.x e. A ps))

Syntax hints:   -> wi 2   <-> wb 127   e. wcel 1092  A.wral 1201

This theorem is referenced by:  isowe 2941  rankonid 3538  kmlem12 3591  kmlem13 3592

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

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