Theorem r19.3rmv 3417

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 1489	. 2 |- F/ x ph
2	1	r19.3rm 3415	1 |- ( E. y y e. A -> ( ph <-> A. x e. A ph ) )

Syntax hints:   -> wi 4   <-> wb 104   E.wex 1449   e. wcel 1461   A.wral 2388

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-nf 1418  df-cleq 2106  df-clel 2109  df-ral 2393

This theorem is referenced by:  iinconstm  3786  exmidsssnc  4084  cnvpom  5037  ssfilem  6720  diffitest  6732  inffiexmid  6751  ctssexmid  6972  caucvgsrlemasr  7526  resqrexlemgt0  10678