Theorem r19.9rmv 3401

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 2162	. . 3 |- ( a = y -> ( a e. A <-> y e. A ) )
2	1	cbvexv 1855	. 2 |- ( E. a a e. A <-> E. y y e. A )
3		eleq1 2162	. . . 4 |- ( a = x -> ( a e. A <-> x e. A ) )
4	3	cbvexv 1855	. . 3 |- ( E. a a e. A <-> E. x x e. A )
5		df-rex 2381	. . . . 5 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
6		19.41v 1841	. . . . 5 |- ( E. x ( x e. A /\ ph ) <-> ( E. x x e. A /\ ph ) )
7	5, 6	bitri 183	. . . 4 |- ( E. x e. A ph <-> ( E. x x e. A /\ ph ) )
8	7	baibr 873	. . 3 |- ( E. x x e. A -> ( ph <-> E. x e. A ph ) )
9	4, 8	sylbi 120	. 2 |- ( E. a a e. A -> ( ph <-> E. x e. A ph ) )
10	2, 9	sylbir 134	1 |- ( E. y y e. A -> ( ph <-> E. x e. A ph ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   E.wex 1436   e. wcel 1448   E.wrex 2376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-cleq 2093  df-clel 2096  df-rex 2381

This theorem is referenced by:  r19.45mv  3403  r19.44mv  3404  iunconstm  3768  fconstfvm  5570  frecabcl  6226  ltexprlemloc  7316  lcmgcdlem  11551