Theorem r19.28av 1755

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:   ph,x

Proof of Theorem r19.28av

Step	Hyp	Ref	Expression
1		r19.27av 1754	. 2 |- ((A.x e. A ps /\ ph) -> A.x e. A (ps /\ ph))
2		ancom 435	. 2 |- ((ph /\ A.x e. A ps) <-> (A.x e. A ps /\ ph))
3		ancom 435	. . 3 |- ((ph /\ ps) <-> (ps /\ ph))
4	3	ralbii 1667	. 2 |- (A.x e. A (ph /\ ps) <-> A.x e. A (ps /\ ph))
5	1, 2, 4	3imtr4 219	1 |- ((ph /\ A.x e. A ps) -> A.x e. A (ph /\ ps))

Syntax hints:   -> wi 3   /\ wa 223  A.wral 1645

This theorem is referenced by:  fununi 3563  fsummulc1 7033  minveclem27 8571

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975

This theorem depends on definitions:  df-bi 147  df-an 225  df-ral 1649

Copyright terms: Public domain