Theorem hbral 1236

Description: Bound-variable hypothesis builder for restricted quantification.

Hypotheses

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

Assertion

Ref	Expression
hbral	|- (A.y e. A ph -> A.xA.y e. A ph)

Distinct variable group(s):   x,y

Proof of Theorem hbral

Step	Hyp	Ref	Expression
1		hbral.1	. . . 4 |- (y e. A -> A.x y e. A)
2		hbral.2	. . . 4 |- (ph -> A.xph)
3	1, 2	hbim 702	. . 3 |- ((y e. A -> ph) -> A.x(y e. A -> ph))
4	3	hbal 700	. 2 |- (A.y(y e. A -> ph) -> A.xA.y(y e. A -> ph))
5		df-ral 1205	. 2 |- (A.y e. A ph <-> A.y(y e. A -> ph))
6	5	bial 695	. 2 |- (A.xA.y e. A ph <-> A.xA.y(y e. A -> ph))
7	4, 5, 6	3imtr4 192	1 |- (A.y e. A ph -> A.xA.y e. A ph)

Syntax hints:   -> wi 2  A.wal 672   e. wcel 1092  A.wral 1201

This theorem is referenced by:  tfis 2245  ralxp 2456  f1fvf 2917  hbiso 2930  isotrALT 2936  hbrdg 2974  scottexs 3543  scott0s 3544  ac6lem 3575  hta 3619

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-ral 1205

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