Theorem r19.28av 2625

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 2624	. 2 |- ( ( A. x e. A ps /\ ph ) -> A. x e. A ( ps /\ ph ) )
2		ancom 266	. 2 |- ( ( ph /\ A. x e. A ps ) <-> ( A. x e. A ps /\ ph ) )
3		ancom 266	. . 3 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
4	3	ralbii 2495	. 2 |- ( A. x e. A ( ph /\ ps ) <-> A. x e. A ( ps /\ ph ) )
5	1, 2, 4	3imtr4i 201	1 |- ( ( ph /\ A. x e. A ps ) -> A. x e. A ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wral 2467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-4 1520  ax-17 1536

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-ral 2472

This theorem is referenced by:  rr19.28v  2891  fununi  5298