Theorem r19.41 2764

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/xps

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 630	. . . 4 |- (((x e. A /\ ph) /\ ps) <-> (x e. A /\ (ph /\ ps)))
2	1	exbii 1582	. . 3 |- (E.x((x e. A /\ ph) /\ ps) <-> E.x(x e. A /\ (ph /\ ps)))
3		r19.41.1	. . . 4 |- F/xps
4	3	19.41 1879	. . 3 |- (E.x((x e. A /\ ph) /\ ps) <-> (E.x(x e. A /\ ph) /\ ps))
5	2, 4	bitr3i 242	. 2 |- (E.x(x e. A /\ (ph /\ ps)) <-> (E.x(x e. A /\ ph) /\ ps))
6		df-rex 2621	. 2 |- (E.x e. A (ph /\ ps) <-> E.x(x e. A /\ (ph /\ ps)))
7		df-rex 2621	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
8	7	anbi1i 676	. 2 |- ((E.x e. A ph /\ ps) <-> (E.x(x e. A /\ ph) /\ ps))
9	5, 6, 8	3bitr4i 268	1 |- (E.x e. A (ph /\ ps) <-> (E.x e. A ph /\ ps))

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   F/wnf 1544   e. wcel 1710  E.wrex 2616

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-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-rex 2621

This theorem is referenced by:  r19.41v  2765