Theorem r19.3rmv 3551

Description: Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 6-Aug-2018.)

Assertion

Ref	Expression
r19.3rmv	|- ( E. y y e. A -> ( ph <-> A. x e. A ph ) )

Distinct variable groups:   x, A   y, A   ph, x

Allowed substitution hint:   ph( y)

Proof of Theorem r19.3rmv

Step	Hyp	Ref	Expression
1		nfv 1551	. 2 |- F/ x ph
2	1	r19.3rm 3549	1 |- ( E. y y e. A -> ( ph <-> A. x e. A ph ) )

Syntax hints:   -> wi 4   <-> wb 105   E.wex 1515   e. wcel 2176   A.wral 2484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-cleq 2198  df-clel 2201  df-ral 2489

This theorem is referenced by:  iinconstm  3936  exmidsssnc  4247  cnvpom  5225  ssfilem  6972  diffitest  6984  inffiexmid  7003  ctssexmid  7252  exmidonfinlem  7301  caucvgsrlemasr  7903  resqrexlemgt0  11331  rmodislmodlem  14112  rmodislmod  14113