Theorem bnj1441 30911

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1441.1	|- ( x e. A -> A. y x e. A )
bnj1441.2	|- ( ph -> A. y ph )

Assertion

Ref	Expression
bnj1441	|- ( z e. { x e. A | ph } -> A. y z e. { x e. A | ph } )

Distinct variable group:   y, z

Allowed substitution hints:   ph( x, y, z)   A( x, y, z)

Proof of Theorem bnj1441

Step	Hyp	Ref	Expression
1		df-rab 2921	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
2		bnj1441.1	. . . 4 |- ( x e. A -> A. y x e. A )
3		bnj1441.2	. . . 4 |- ( ph -> A. y ph )
4	2, 3	hban 2128	. . 3 |- ( ( x e. A /\ ph ) -> A. y ( x e. A /\ ph ) )
5	4	hbab 2613	. 2 |- ( z e. { x | ( x e. A /\ ph ) } -> A. y z e. { x | ( x e. A /\ ph ) } )
6	1, 5	hbxfreq 2730	1 |- ( z e. { x e. A | ph } -> A. y z e. { x e. A | ph } )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   e. wcel 1990   {cab 2608   {crab 2916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-rab 2921

This theorem is referenced by: (None)