Theorem r19.9rzv 1768

Description: Restricted quantification of wff not containing quantified variable.

Assertion

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

Distinct variable group(s):   x,A   ph,x

Proof of Theorem r19.9rzv

Step	Hyp	Ref	Expression
1		r19.3rzv 1767	. . . 4 |- (-. A = (/) -> (-. ph <-> A.x e. A -. ph))
2	1	bicomd 399	. . 3 |- (-. A = (/) -> (A.x e. A -. ph <-> -. ph))
3	2	bicon2d 404	. 2 |- (-. A = (/) -> (ph <-> -. A.x e. A -. ph))
4		dfrex2 1212	. 2 |- (E.x e. A ph <-> -. A.x e. A -. ph)
5	3, 4	syl6bbr 416	1 |- (-. A = (/) -> (ph <-> E.x e. A ph))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   = wceq 1091  A.wral 1201  E.wrex 1202  (/)c0 1707

This theorem is referenced by:  r19.45zv 1770  r19.36zv 1772  fconstfv 2903

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-nul 1708

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