Theorem hbrex 1238

Description: Bound-variable hypothesis builder for restricted quantification.

Hypotheses

Ref	Expression
hbrex.1	|- (y e. A -> A.x y e. A)
hbrex.2	|- (ph -> A.xph)

Assertion

Ref	Expression
hbrex	|- (E.y e. A ph -> A.xE.y e. A ph)

Distinct variable group(s):   x,y

Proof of Theorem hbrex

Step	Hyp	Ref	Expression
1		hbrex.1	. . . 4 |- (y e. A -> A.x y e. A)
2		hbrex.2	. . . 4 |- (ph -> A.xph)
3	1, 2	hban 704	. . 3 |- ((y e. A /\ ph) -> A.x(y e. A /\ ph))
4	3	hbex 701	. 2 |- (E.y(y e. A /\ ph) -> A.xE.y(y e. A /\ ph))
5		df-rex 1206	. 2 |- (E.y e. A ph <-> E.y(y e. A /\ ph))
6	5	bial 695	. 2 |- (A.xE.y e. A ph <-> A.xE.y(y e. A /\ ph))
7	4, 5, 6	3imtr4 192	1 |- (E.y e. A ph -> A.xE.y e. A ph)

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

This theorem is referenced by:  r19.12 1281  iunrab 2022  abrexexlem2 2911  abrexex2 2915  hbrdg 2974  elrnoprab 3054

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-7 676  ax-gen 677

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

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