Theorem hbral 2395

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 2353	. 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 1477	. . 3 |- ( ( y e. A -> ph ) -> A. x ( y e. A -> ph ) )
5	4	hbal 1406	. 2 |- ( A. y ( y e. A -> ph ) -> A. x A. y ( y e. A -> ph ) )
6	1, 5	hbxfrbi 1401	1 |- ( A. y e. A ph -> A. x A. y e. A ph )

Syntax hints:   -> wi 4   A.wal 1282   e. wcel 1433   A.wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-4 1440  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-ral 2353

This theorem is referenced by: (None)