Theorem r19.3rm 3310

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 2100	. . 3 |- ( a = y -> ( a e. A <-> y e. A ) )
2	1	cbvexv 1795	. 2 |- ( E. a a e. A <-> E. y y e. A )
3		eleq1 2100	. . . 4 |- ( a = x -> ( a e. A <-> x e. A ) )
4	3	cbvexv 1795	. . 3 |- ( E. a a e. A <-> E. x x e. A )
5		biimt 230	. . . 4 |- ( E. x x e. A -> ( ph <-> ( E. x x e. A -> ph ) ) )
6		df-ral 2311	. . . . 5 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
7		r19.3rm.1	. . . . . 6 |- F/ x ph
8	7	19.23 1568	. . . . 5 |- ( A. x ( x e. A -> ph ) <-> ( E. x x e. A -> ph ) )
9	6, 8	bitri 173	. . . 4 |- ( A. x e. A ph <-> ( E. x x e. A -> ph ) )
10	5, 9	syl6bbr 187	. . 3 |- ( E. x x e. A -> ( ph <-> A. x e. A ph ) )
11	4, 10	sylbi 114	. 2 |- ( E. a a e. A -> ( ph <-> A. x e. A ph ) )
12	2, 11	sylbir 125	1 |- ( E. y y e. A -> ( ph <-> A. x e. A ph ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   F/wnf 1349   E.wex 1381   e. wcel 1393   A.wral 2306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-cleq 2033  df-clel 2036  df-ral 2311

This theorem is referenced by:  r19.28m  3311  r19.3rmv  3312  r19.27m  3316  indstr  8536