Theorem r19.9rmv 3333

Description: Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.)

Assertion

Ref	Expression
r19.9rmv	|- ( E. y y e. A -> ( ph <-> E. x e. A ph ) )

Distinct variable groups:   x, A   y, A   ph, x

Allowed substitution hint:   ph( y)

Proof of Theorem r19.9rmv

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2141	. . 3 |- ( a = y -> ( a e. A <-> y e. A ) )
2	1	cbvexv 1836	. 2 |- ( E. a a e. A <-> E. y y e. A )
3		eleq1 2141	. . . 4 |- ( a = x -> ( a e. A <-> x e. A ) )
4	3	cbvexv 1836	. . 3 |- ( E. a a e. A <-> E. x x e. A )
5		df-rex 2354	. . . . 5 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
6		19.41v 1823	. . . . 5 |- ( E. x ( x e. A /\ ph ) <-> ( E. x x e. A /\ ph ) )
7	5, 6	bitri 182	. . . 4 |- ( E. x e. A ph <-> ( E. x x e. A /\ ph ) )
8	7	baibr 862	. . 3 |- ( E. x x e. A -> ( ph <-> E. x e. A ph ) )
9	4, 8	sylbi 119	. 2 |- ( E. a a e. A -> ( ph <-> E. x e. A ph ) )
10	2, 9	sylbir 133	1 |- ( E. y y e. A -> ( ph <-> E. x e. A ph ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   E.wex 1421   e. wcel 1433   E.wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077  df-rex 2354

This theorem is referenced by:  r19.45mv  3335  iunconstm  3686  fconstfvm  5400  ltexprlemloc  6797  lcmgcdlem  10459