Theorem r19.23advv 1288

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

Hypothesis

Ref	Expression
r19.23advv.1	|- (ph -> ((x e. A /\ y e. B) -> (ps -> ch)))

Assertion

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

Distinct variable group(s):   x,y,ph   ch,x,y   y,A

Proof of Theorem r19.23advv

Step	Hyp	Ref	Expression
1		r19.23advv.1	. . . . . 6 |- (ph -> ((x e. A /\ y e. B) -> (ps -> ch)))
2	1	exp3a 292	. . . . 5 |- (ph -> (x e. A -> (y e. B -> (ps -> ch))))
3	2	imp 277	. . . 4 |- ((ph /\ x e. A) -> (y e. B -> (ps -> ch)))
4	3	r19.23adv 1286	. . 3 |- ((ph /\ x e. A) -> (E.y e. B ps -> ch))
5	4	exp 291	. 2 |- (ph -> (x e. A -> (E.y e. B ps -> ch)))
6	5	r19.23adv 1286	1 |- (ph -> (E.x e. A E.y e. B ps -> ch))

Syntax hints:   -> wi 2   /\ wa 196   e. wcel 1092  E.wrex 1202

This theorem is referenced by:  xpdom2 3345  unxpdomlem 3649  qaddclt 4642  qmulclt 4644  xpnnen 4927  infxpidmlem7 4939  shselt 5280  shmods 5363  sumdmdlem 5786

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-an 198  df-ex 679  df-rex 1206

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