Theorem r19.3rm 3540

Description: Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.)

Hypothesis

Ref	Expression
r19.3rm.1	|- F/ x ph

Assertion

Ref	Expression
r19.3rm	|- ( E. y y e. A -> ( ph <-> A. x e. A ph ) )

Distinct variable groups:   x, A   y, A

Allowed substitution hints:   ph( x, y)

Proof of Theorem r19.3rm

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2259	. . 3 |- ( a = y -> ( a e. A <-> y e. A ) )
2	1	cbvexv 1933	. 2 |- ( E. a a e. A <-> E. y y e. A )
3		eleq1 2259	. . . 4 |- ( a = x -> ( a e. A <-> x e. A ) )
4	3	cbvexv 1933	. . 3 |- ( E. a a e. A <-> E. x x e. A )
5		biimt 241	. . . 4 |- ( E. x x e. A -> ( ph <-> ( E. x x e. A -> ph ) ) )
6		df-ral 2480	. . . . 5 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
7		r19.3rm.1	. . . . . 6 |- F/ x ph
8	7	19.23 1692	. . . . 5 |- ( A. x ( x e. A -> ph ) <-> ( E. x x e. A -> ph ) )
9	6, 8	bitri 184	. . . 4 |- ( A. x e. A ph <-> ( E. x x e. A -> ph ) )
10	5, 9	bitr4di 198	. . 3 |- ( E. x x e. A -> ( ph <-> A. x e. A ph ) )
11	4, 10	sylbi 121	. 2 |- ( E. a a e. A -> ( ph <-> A. x e. A ph ) )
12	2, 11	sylbir 135	1 |- ( E. y y e. A -> ( ph <-> A. x e. A ph ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   F/wnf 1474   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:  r19.28m  3541  r19.3rmv  3542  r19.27m  3547  indstr  9684