Theorem r19.9rzv 4065

Description: Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998.)

Assertion

Ref	Expression
r19.9rzv	|- ( A =/= (/) -> ( ph <-> E. x e. A ph ) )

Distinct variable groups:   x, A   ph, x

Proof of Theorem r19.9rzv

Step	Hyp	Ref	Expression
1		dfrex2 2996	. 2 |- ( E. x e. A ph <-> -. A. x e. A -. ph )
2		r19.3rzv 4064	. . 3 |- ( A =/= (/) -> ( -. ph <-> A. x e. A -. ph ) )
3	2	con1bid 345	. 2 |- ( A =/= (/) -> ( -. A. x e. A -. ph <-> ph ) )
4	1, 3	syl5rbb 273	1 |- ( A =/= (/) -> ( ph <-> E. x e. A ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   =/= wne 2794   A.wral 2912   E.wrex 2913   (/)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-rex 2918  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  r19.45zv  4068  r19.44zv  4069  r19.36zv  4072  iunconst  4529  lcmgcdlem  15319  pmtrprfvalrn  17908  dvdsr02  18656  voliune  30292  dya2iocuni  30345  filnetlem4  32376  prmunb2  38510