Theorem r19.28v 2560

Description: Restricted quantifier version of one direction of 19.28 1542. (The other direction holds when A is inhabited, see r19.28mv 3455.) (Contributed by NM, 2-Apr-2004.) (Proof shortened by Wolf Lammen, 17-Jun-2023.)

Assertion

Ref	Expression
r19.28v	|- ( ( 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 r19.28v

Step	Hyp	Ref	Expression
1		id 19	. . . 4 |- ( ph -> ph )
2	1	ralrimivw 2506	. . 3 |- ( ph -> A. x e. A ph )
3	2	anim1i 338	. 2 |- ( ( ph /\ A. x e. A ps ) -> ( A. x e. A ph /\ A. x e. A ps ) )
4		r19.26 2558	. 2 |- ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) )
5	3, 4	sylibr 133	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-nf 1437  df-ral 2421

This theorem is referenced by:  txlm  12448