Theorem r19.3rzv 2338

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 2278	. . 3 |- (A =/= (/) <-> E.x x e. A)
2		biimt 729	. . 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 1641	. . 3 |- (A.x e. A ph <-> A.x(x e. A -> ph))
5		19.23v 1288	. . 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 536	1 |- (A =/= (/) -> (ph <-> A.x e. A ph))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 951   e. wcel 955  E.wex 977   =/= wne 1577  A.wral 1637  (/)c0 2270

This theorem is referenced by:  r19.9rzv 2339  r19.28zv 2340  r19.37zv 2341  r19.27zv 2343  iin0 2730  cnvpo 3508  fint 3635

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-v 1803  df-dif 2039  df-nul 2271

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