Theorem r19.28av 2494

Description: Restricted version of one direction of Theorem 19.28 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by NM, 2-Apr-2004.)

Assertion

Ref	Expression
r19.28av	|- ( ( 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 r19.28av

Step	Hyp	Ref	Expression
1		r19.27av 2493	. 2 |- ( ( A. x e. A ps /\ ph ) -> A. x e. A ( ps /\ ph ) )
2		ancom 262	. 2 |- ( ( ph /\ A. x e. A ps ) <-> ( A. x e. A ps /\ ph ) )
3		ancom 262	. . 3 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
4	3	ralbii 2373	. 2 |- ( A. x e. A ( ph /\ ps ) <-> A. x e. A ( ps /\ ph ) )
5	1, 2, 4	3imtr4i 199	1 |- ( ( ph /\ A. x e. A ps ) -> A. x e. A ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 102   A.wral 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-4 1441  ax-17 1460

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-ral 2354

This theorem is referenced by:  rr19.28v  2735  fununi  4998