Theorem r19.41 2625

Description: Restricted quantifier version of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 1-Nov-2010.)

Hypothesis

Ref	Expression
r19.41.1	|- F/ x ps

Assertion

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

Proof of Theorem r19.41

Step	Hyp	Ref	Expression
1		anass 399	. . . 4 |- ( ( ( x e. A /\ ph ) /\ ps ) <-> ( x e. A /\ ( ph /\ ps ) ) )
2	1	exbii 1598	. . 3 |- ( E. x ( ( x e. A /\ ph ) /\ ps ) <-> E. x ( x e. A /\ ( ph /\ ps ) ) )
3		r19.41.1	. . . 4 |- F/ x ps
4	3	19.41 1679	. . 3 |- ( E. x ( ( x e. A /\ ph ) /\ ps ) <-> ( E. x ( x e. A /\ ph ) /\ ps ) )
5	2, 4	bitr3i 185	. 2 |- ( E. x ( x e. A /\ ( ph /\ ps ) ) <-> ( E. x ( x e. A /\ ph ) /\ ps ) )
6		df-rex 2454	. 2 |- ( E. x e. A ( ph /\ ps ) <-> E. x ( x e. A /\ ( ph /\ ps ) ) )
7		df-rex 2454	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
8	7	anbi1i 455	. 2 |- ( ( E. x e. A ph /\ ps ) <-> ( E. x ( x e. A /\ ph ) /\ ps ) )
9	5, 6, 8	3bitr4i 211	1 |- ( E. x e. A ( ph /\ ps ) <-> ( E. x e. A ph /\ ps ) )

Syntax hints:   /\ wa 103   <-> wb 104   F/wnf 1453   E.wex 1485   e. wcel 2141   E.wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-rex 2454

This theorem is referenced by:  r19.41v  2626