Theorem r19.28av 2754

Description: Restricted version of one direction of Theorem 19.28 of [Margaris] p. 90. (The other direction doesn't hold when A is empty.) (Contributed by NM, 2-Apr-2004.)

Assertion

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

Step	Hyp	Ref	Expression
1		r19.27av 2753	. 2 |- ((A.x e. A ps /\ ph) -> A.x e. A (ps /\ ph))
2		ancom 437	. 2 |- ((ph /\ A.x e. A ps) <-> (A.x e. A ps /\ ph))
3		ancom 437	. . 3 |- ((ph /\ ps) <-> (ps /\ ph))
4	3	ralbii 2639	. 2 |- (A.x e. A (ph /\ ps) <-> A.x e. A (ps /\ ph))
5	1, 2, 4	3imtr4i 257	1 |- ((ph /\ A.x e. A ps) -> A.x e. A (ph /\ ps))

Syntax hints:   -> wi 4   /\ wa 358  A.wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by:  rr19.28v  2982  fununi  5161