Theorem r19.3rzv 2352

Description: Restricted quantification of wff not containing quantified variable.

Assertion

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

Distinct variable groups:   x,A   ph,x

Proof of Theorem r19.3rzv

Step	Hyp	Ref	Expression
1		ne0 2292	. . 3 |- (A =/= (/) <-> E.x x e. A)
2		biimt 733	. . 3 |- (E.x x e. A -> (ph <-> (E.x x e. A -> ph)))
3	1, 2	sylbi 199	. 2 |- (A =/= (/) -> (ph <-> (E.x x e. A -> ph)))
4		df-ral 1652	. . 3 |- (A.x e. A ph <-> A.x(x e. A -> ph))
5		19.23v 1295	. . 3 |- (A.x(x e. A -> ph) <-> (E.x x e. A -> ph))
6	4, 5	bitr 173	. 2 |- (A.x e. A ph <-> (E.x x e. A -> ph))
7	3, 6	syl6bbr 540	1 |- (A =/= (/) -> (ph <-> A.x e. A ph))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 956   e. wcel 960  E.wex 982   =/= wne 1588  A.wral 1648  (/)c0 2283

This theorem is referenced by:  r19.9rzv 2353  r19.28zv 2354  r19.37zv 2355  r19.27zv 2357  iin0 2745  cnvpo 3528  fint 3656

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-dif 2052  df-nul 2284

Copyright terms: Public domain