Theorem r19.23ad 1285

Description: Deduction from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.)

Hypotheses

Ref	Expression
r19.23ad.1	|- (ph -> A.xph)
r19.23ad.2	|- (ch -> A.xch)
r19.23ad.3	|- (ph -> (x e. A -> (ps -> ch)))

Assertion

Ref	Expression
r19.23ad	|- (ph -> (E.x e. A ps -> ch))

Proof of Theorem r19.23ad

Step	Hyp	Ref	Expression
1		r19.23ad.1	. . 3 |- (ph -> A.xph)
2		r19.23ad.2	. . 3 |- (ch -> A.xch)
3		r19.23ad.3	. . . 4 |- (ph -> (x e. A -> (ps -> ch)))
4	3	imp3a 279	. . 3 |- (ph -> ((x e. A /\ ps) -> ch))
5	1, 2, 4	19.23ad 748	. 2 |- (ph -> (E.x(x e. A /\ ps) -> ch))
6		df-rex 1206	. 2 |- (E.x e. A ps <-> E.x(x e. A /\ ps))
7	5, 6	syl5ib 181	1 |- (ph -> (E.x e. A ps -> ch))

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672  E.wex 678   e. wcel 1092  E.wrex 1202

This theorem is referenced by:  reuuni4 1959  onfr 2237  peano5 2394  ffnfv 2892  iunon 2947  iinon 2948  tz7.49 2997  nneneq 3408  zornlem4 3606  zornlem5 3607  atom1d 5750

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

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

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