Theorem r19.28av 2568

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 2567	. 2 |- ( ( A. x e. A ps /\ ph ) -> A. x e. A ( ps /\ ph ) )
2		ancom 264	. 2 |- ( ( ph /\ A. x e. A ps ) <-> ( A. x e. A ps /\ ph ) )
3		ancom 264	. . 3 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
4	3	ralbii 2441	. 2 |- ( A. x e. A ( ph /\ ps ) <-> A. x e. A ( ps /\ ph ) )
5	1, 2, 4	3imtr4i 200	1 |- ( ( ph /\ A. x e. A ps ) -> A. x e. A ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wral 2416

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-5 1423  ax-gen 1425  ax-4 1487  ax-17 1506

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-ral 2421

This theorem is referenced by:  rr19.28v  2824  fununi  5191