Theorem r19.3rmv 3537

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 1539	. 2 |- F/ x ph
2	1	r19.3rm 3535	1 |- ( E. y y e. A -> ( ph <-> A. x e. A ph ) )

Syntax hints:   -> wi 4   <-> wb 105   E.wex 1503   e. wcel 2164   A.wral 2472

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-cleq 2186  df-clel 2189  df-ral 2477

This theorem is referenced by:  iinconstm  3921  exmidsssnc  4232  cnvpom  5208  ssfilem  6931  diffitest  6943  inffiexmid  6962  ctssexmid  7209  exmidonfinlem  7253  caucvgsrlemasr  7850  resqrexlemgt0  11164  rmodislmodlem  13846  rmodislmod  13847