Theorem r19.9rmv 3454

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 2202	. . 3 |- ( a = y -> ( a e. A <-> y e. A ) )
2	1	cbvexv 1890	. 2 |- ( E. a a e. A <-> E. y y e. A )
3		eleq1 2202	. . . 4 |- ( a = x -> ( a e. A <-> x e. A ) )
4	3	cbvexv 1890	. . 3 |- ( E. a a e. A <-> E. x x e. A )
5		df-rex 2422	. . . . 5 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
6		19.41v 1874	. . . . 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 905	. . 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 1468   e. wcel 1480   E.wrex 2417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-clel 2135  df-rex 2422

This theorem is referenced by:  r19.45mv  3456  r19.44mv  3457  iunconstm  3821  fconstfvm  5638  frecabcl  6296  ltexprlemloc  7415  lcmgcdlem  11758