Theorem r19.28v 3071

Description: Restricted quantifier version of one direction of 19.28 2096. (The other direction holds when A is nonempty, see r19.28zv 4066.) (Contributed by NM, 2-Apr-2004.)

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		r19.27v 3070	. 2 |- ( ( A. x e. A ps /\ ph ) -> A. x e. A ( ps /\ ph ) )
2		ancom 466	. 2 |- ( ( ph /\ A. x e. A ps ) <-> ( A. x e. A ps /\ ph ) )
3		ancom 466	. . 3 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
4	3	ralbii 2980	. 2 |- ( A. x e. A ( ph /\ ps ) <-> A. x e. A ( ps /\ ph ) )
5	1, 2, 4	3imtr4i 281	1 |- ( ( ph /\ A. x e. A ps ) -> A. x e. A ( ph /\ ps ) )

Syntax hints:   -> wi 4   /\ wa 384   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:  rr19.28v  3346  fununi  5964  txlm  21451