Theorem r19.3rz 4062

Description: Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.)

Hypothesis

Ref	Expression
r19.3rz.1	|- F/ x ph

Assertion

Ref	Expression
r19.3rz	|- ( A =/= (/) -> ( ph <-> A. x e. A ph ) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem r19.3rz

Step	Hyp	Ref	Expression
1		n0 3931	. . 3 |- ( A =/= (/) <-> E. x x e. A )
2		biimt 350	. . 3 |- ( E. x x e. A -> ( ph <-> ( E. x x e. A -> ph ) ) )
3	1, 2	sylbi 207	. 2 |- ( A =/= (/) -> ( ph <-> ( E. x x e. A -> ph ) ) )
4		df-ral 2917	. . 3 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
5		r19.3rz.1	. . . 4 |- F/ x ph
6	5	19.23 2080	. . 3 |- ( A. x ( x e. A -> ph ) <-> ( E. x x e. A -> ph ) )
7	4, 6	bitri 264	. 2 |- ( A. x e. A ph <-> ( E. x x e. A -> ph ) )
8	3, 7	syl6bbr 278	1 |- ( A =/= (/) -> ( ph <-> A. x e. A ph ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704   F/wnf 1708   e. wcel 1990   =/= wne 2794   A.wral 2912   (/)c0 3915

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-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  r19.28z  4063  r19.3rzv  4064  r19.27z  4070  2reu4a  41189