Theorem r19.32vr 2577

Description: One direction of Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. For decidable propositions this is an equivalence, as seen at r19.32vdc 2578. (Contributed by Jim Kingdon, 19-Aug-2018.)

Assertion

Ref	Expression
r19.32vr	|- ( ( ph \/ A. x e. A ps ) -> A. x e. A ( ph \/ ps ) )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   A( x)

Proof of Theorem r19.32vr

Step	Hyp	Ref	Expression
1		nfv 1508	. 2 |- F/ x ph
2	1	r19.32r 2576	1 |- ( ( ph \/ A. x e. A ps ) -> A. x e. A ( ph \/ ps ) )

Syntax hints:   -> wi 4   \/ wo 697   A.wral 2414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-gen 1425  ax-4 1487  ax-17 1506

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-ral 2419

This theorem is referenced by:  iinuniss  3890