Theorem r19.3rmv 3587

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 1577	. 2 |- F/ x ph
2	1	r19.3rm 3585	1 |- ( E. y y e. A -> ( ph <-> A. x e. A ph ) )

Syntax hints:   -> wi 4   <-> wb 105   E.wex 1541   e. wcel 2202   A.wral 2511

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-cleq 2224  df-clel 2227  df-ral 2516

This theorem is referenced by:  iinconstm  3984  exmidsssnc  4299  cnvpom  5286  ssfilem  7105  ssfilemd  7107  diffitest  7119  inffiexmid  7141  ctssexmid  7409  exmidonfinlem  7464  caucvgsrlemasr  8070  resqrexlemgt0  11660  rmodislmodlem  14446  rmodislmod  14447