Theorem r19.9rmv 3619

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 2301	. . 3 |- ( a = y -> ( a e. A <-> y e. A ) )
2	1	cbvexv 1974	. 2 |- ( E. a a e. A <-> E. y y e. A )
3		eleq1 2301	. . . 4 |- ( a = x -> ( a e. A <-> x e. A ) )
4	3	cbvexv 1974	. . 3 |- ( E. a a e. A <-> E. x x e. A )
5		df-rex 2534	. . . . 5 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
6		19.41v 1958	. . . . 5 |- ( E. x ( x e. A /\ ph ) <-> ( E. x x e. A /\ ph ) )
7	5, 6	bitri 184	. . . 4 |- ( E. x e. A ph <-> ( E. x x e. A /\ ph ) )
8	7	baibr 932	. . 3 |- ( E. x x e. A -> ( ph <-> E. x e. A ph ) )
9	4, 8	sylbi 121	. 2 |- ( E. a a e. A -> ( ph <-> E. x e. A ph ) )
10	2, 9	sylbir 135	1 |- ( E. y y e. A -> ( ph <-> E. x e. A ph ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   E.wex 1545   e. wcel 2209   E.wrex 2529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234  df-rex 2534

This theorem is used by:  r19.45mv  3621  r19.44mv  3622  iunconstm  4020  fconstfvm  5933  frecabcl  6670  ltexprlemloc  7974  lcmgcdlem  12855  dvdsr02  14412