Theorem r19.27v 3070

Description: Restricted quantitifer version of one direction of 19.27 2095. (The other direction holds when A is nonempty, see r19.27zv 4071.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Assertion

Ref	Expression
r19.27v	|- ( ( 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 r19.27v

Step	Hyp	Ref	Expression
1		ax-1 6	. . . 4 |- ( ps -> ( x e. A -> ps ) )
2	1	ralrimiv 2965	. . 3 |- ( ps -> A. x e. A ps )
3	2	anim2i 593	. 2 |- ( ( A. x e. A ph /\ ps ) -> ( A. x e. A ph /\ A. x e. A ps ) )
4		r19.26 3064	. 2 |- ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) )
5	3, 4	sylibr 224	1 |- ( ( A. x e. A ph /\ ps ) -> A. x e. A ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839

This theorem depends on definitions:  df-bi 197  df-an 386  df-ral 2917

This theorem is referenced by:  r19.28v  3071  txlm  21451  tx1stc  21453  spanuni  28403