Theorem r19.3rmv 3542

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 1542	. 2 |- F/ x ph
2	1	r19.3rm 3540	1 |- ( E. y y e. A -> ( ph <-> A. x e. A ph ) )

Syntax hints:   -> wi 4   <-> wb 105   E.wex 1506   e. wcel 2167   A.wral 2475

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-cleq 2189  df-clel 2192  df-ral 2480

This theorem is referenced by:  iinconstm  3926  exmidsssnc  4237  cnvpom  5213  ssfilem  6945  diffitest  6957  inffiexmid  6976  ctssexmid  7225  exmidonfinlem  7272  caucvgsrlemasr  7874  resqrexlemgt0  11202  rmodislmodlem  13982  rmodislmod  13983