Theorem hbral 2783

Description: Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999.) (Revised by David Abernethy, 13-Dec-2009.)

Hypotheses

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

Assertion

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

Proof of Theorem hbral

Step	Hyp	Ref	Expression
1		df-ral 2753	. 2 |- ( A. y e. A ph <-> A. y ( y e. A -> ph ) )
2		hbral.1	. . . 4 |- ( y e. A -> A. x y e. A )
3		hbral.2	. . . 4 |- ( ph -> A. x ph )
4	2, 3	hbim 2015	. . 3 |- ( ( y e. A -> ph ) -> A. x ( y e. A -> ph ) )
5	4	hbal 1932	. 2 |- ( A. y ( y e. A -> ph ) -> A. x A. y ( y e. A -> ph ) )
6	1, 5	hbxfrbi 1704	1 |- ( A. y e. A ph -> A. x A. y e. A ph )

Syntax hints:   -> wi 4   A.wal 1452   e. wcel 1897   A.wral 2748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943

This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678  df-ral 2753

This theorem is referenced by:  tratrbVD  37297