Theorem r19.27av 2622

Description: Restricted version of one direction of Theorem 19.27 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Assertion

Ref	Expression
r19.27av	|- ( ( A. x e. A ph /\ ps ) -> A. x e. A ( ph /\ ps ) )

Distinct variable group:   ps, x

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

Proof of Theorem r19.27av

Step	Hyp	Ref	Expression
1		ax-1 6	. . . 4 |- ( ps -> ( x e. A -> ps ) )
2	1	ralrimiv 2559	. . 3 |- ( ps -> A. x e. A ps )
3	2	anim2i 342	. 2 |- ( ( A. x e. A ph /\ ps ) -> ( A. x e. A ph /\ A. x e. A ps ) )
4		r19.26 2613	. 2 |- ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) )
5	3, 4	sylibr 134	1 |- ( ( A. x e. A ph /\ ps ) -> A. x e. A ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2158   A.wral 2465

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-nf 1471  df-ral 2470

This theorem is referenced by:  r19.28av  2623  fimaxre2  11249