Theorem r19.3rmv 3499

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 1516	. 2 |- F/ x ph
2	1	r19.3rm 3497	1 |- ( E. y y e. A -> ( ph <-> A. x e. A ph ) )

Syntax hints:   -> wi 4   <-> wb 104   E.wex 1480   e. wcel 2136   A.wral 2444

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-cleq 2158  df-clel 2161  df-ral 2449

This theorem is referenced by:  iinconstm  3875  exmidsssnc  4182  cnvpom  5146  ssfilem  6841  diffitest  6853  inffiexmid  6872  ctssexmid  7114  exmidonfinlem  7149  caucvgsrlemasr  7731  resqrexlemgt0  10962