Theorem r19.3rmv 3505

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 1521	. 2 |- F/ x ph
2	1	r19.3rm 3503	1 |- ( E. y y e. A -> ( ph <-> A. x e. A ph ) )

Syntax hints:   -> wi 4   <-> wb 104   E.wex 1485   e. wcel 2141   A.wral 2448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-cleq 2163  df-clel 2166  df-ral 2453

This theorem is referenced by:  iinconstm  3882  exmidsssnc  4189  cnvpom  5153  ssfilem  6853  diffitest  6865  inffiexmid  6884  ctssexmid  7126  exmidonfinlem  7170  caucvgsrlemasr  7752  resqrexlemgt0  10984