Theorem r19.3rm 3289

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

Hypothesis

Ref	Expression
r19.3rm.1	|- F/xph

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 2082	. . 3 |- ( a = y -> ( a e. A <-> y e. A))
2	1	cbvexv 1777	. 2 |- (E. a a e. A <-> E.y y e. A)
3		eleq1 2082	. . . 4 |- ( a = x -> ( a e. A <-> x e. A))
4	3	cbvexv 1777	. . 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 2289	. . . . 5 |- (A.x e. A ph <-> A.x(x e. A -> ph))
7		r19.3rm.1	. . . . . 6 |- F/xph
8	7	19.23 1550	. . . . 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 1226   F/wnf 1329  E.wex 1362   e. wcel 1374  A.wral 2284

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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-nf 1330  df-cleq 2015  df-clel 2018  df-ral 2289

This theorem is referenced by:  r19.3rmv  3290