Theorem hbrex 1688

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:   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 1009	. . 3 |- ((y e. A /\ ph) -> A.x(y e. A /\ ph))
4	3	hbex 1006	. 2 |- (E.y(y e. A /\ ph) -> A.xE.y(y e. A /\ ph))
5		df-rex 1650	. 2 |- (E.y e. A ph <-> E.y(y e. A /\ ph))
6	5	albii 999	. 2 |- (A.xE.y e. A ph <-> A.xE.y(y e. A /\ ph))
7	4, 5, 6	3imtr4 219	1 |- (E.y e. A ph -> A.xE.y e. A ph)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   e. wcel 958  E.wex 980  E.wrex 1646

This theorem is referenced by:  r19.12 1740  iunrab 2596  abrexexlem2 3859  abrexex2 3871  hbrdg 3936  elrnoprabg 4124  oarec 4196  hbsum1 6983  hbsum 6984  fgsb 10570  fgsbOLD 10571  fgsb2 10580

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-4 973  ax-5o 975  ax-6o 978

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-rex 1650

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