Theorem r19.41v 1302

Description: Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90.

Assertion

Ref	Expression
r19.41v	|- (E.x e. A (ph /\ ps) <-> (E.x e. A ph /\ ps))

Distinct variable group(s):   ps,x

Proof of Theorem r19.41v

Step	Hyp	Ref	Expression
1		anass 336	. . . 4 |- (((x e. A /\ ph) /\ ps) <-> (x e. A /\ (ph /\ ps)))
2	1	biex 733	. . 3 |- (E.x((x e. A /\ ph) /\ ps) <-> E.x(x e. A /\ (ph /\ ps)))
3		19.41v 963	. . 3 |- (E.x((x e. A /\ ph) /\ ps) <-> (E.x(x e. A /\ ph) /\ ps))
4	2, 3	bitr3 153	. 2 |- (E.x(x e. A /\ (ph /\ ps)) <-> (E.x(x e. A /\ ph) /\ ps))
5		df-rex 1206	. 2 |- (E.x e. A (ph /\ ps) <-> E.x(x e. A /\ (ph /\ ps)))
6		df-rex 1206	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
7	6	anbi1i 368	. 2 |- ((E.x e. A ph /\ ps) <-> (E.x(x e. A /\ ph) /\ ps))
8	4, 5, 7	3bitr4 158	1 |- (E.x e. A (ph /\ ps) <-> (E.x e. A ph /\ ps))

Syntax hints:   <-> wb 127   /\ wa 196  E.wex 678   e. wcel 1092  E.wrex 1202

This theorem is referenced by:  r19.42v 1303  reuxfr 1580  isomin 2937  isoini 2938  mapsnen 3334

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-6 675  ax-gen 677  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-rex 1206

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