Theorem r19.28av 1294

Description: Restricted version of one direction of Theorem 19.28 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.)

Assertion

Ref	Expression
r19.28av	|- ((ph /\ A.x e. A ps) -> A.x e. A (ph /\ ps))

Distinct variable group(s):   ph,x

Proof of Theorem r19.28av

Step	Hyp	Ref	Expression
1		r19.27av 1293	. 2 |- ((A.x e. A ps /\ ph) -> A.x e. A (ps /\ ph))
2		ancom 333	. 2 |- ((ph /\ A.x e. A ps) <-> (A.x e. A ps /\ ph))
3		ancom 333	. . 3 |- ((ph /\ ps) <-> (ps /\ ph))
4	3	biral 1223	. 2 |- (A.x e. A (ph /\ ps) <-> A.x e. A (ps /\ ph))
5	1, 2, 4	3imtr4 192	1 |- ((ph /\ A.x e. A ps) -> A.x e. A (ph /\ ps))

Syntax hints:   -> wi 2   /\ wa 196  A.wral 1201

This theorem is referenced by:  fununi 2705

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ral 1205

metamath.org