Theorem r19.27v 2584

Description: Restricted quantitifer version of one direction of 19.27 1541. (The other direction holds when A is inhabited, see r19.27mv 3490.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)

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		id 19	. . . 4 |- ( ps -> ps )
2	1	ralrimivw 2531	. . 3 |- ( ps -> A. x e. A ps )
3	2	anim2i 340	. 2 |- ( ( A. x e. A ph /\ ps ) -> ( A. x e. A ph /\ A. x e. A ps ) )
4		r19.26 2583	. 2 |- ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) )
5	3, 4	sylibr 133	1 |- ( ( A. x e. A ph /\ ps ) -> A. x e. A ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wral 2435

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 1427  ax-gen 1429  ax-4 1490  ax-17 1506

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-ral 2440

This theorem is referenced by:  txlm  12690