Theorem rgen2a 2417

Description: Generalization rule for restricted quantification. Note that x and y needn't be distinct (and illustrates the use of dvelimor 1935). (Contributed by NM, 23-Nov-1994.) (Proof rewritten by Jim Kingdon, 1-Jun-2018.)

Hypothesis

Ref	Expression
rgen2a.1	|- ( ( x e. A /\ y e. A ) -> ph )

Assertion

Ref	Expression
rgen2a	|- A. x e. A A. y e. A ph

Distinct variable group:   y, A

Allowed substitution hints:   ph( x, y)   A( x)

Proof of Theorem rgen2a

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1461	. . . . 5 |- F/ y z e. A
2		eleq1 2141	. . . . 5 |- ( z = x -> ( z e. A <-> x e. A ) )
3	1, 2	dvelimor 1935	. . . 4 |- ( A. y y = x \/ F/ y x e. A )
4		eleq1 2141	. . . . . . . . 9 |- ( y = x -> ( y e. A <-> x e. A ) )
5		rgen2a.1	. . . . . . . . . 10 |- ( ( x e. A /\ y e. A ) -> ph )
6	5	ex 113	. . . . . . . . 9 |- ( x e. A -> ( y e. A -> ph ) )
7	4, 6	syl6bi 161	. . . . . . . 8 |- ( y = x -> ( y e. A -> ( y e. A -> ph ) ) )
8	7	pm2.43d 49	. . . . . . 7 |- ( y = x -> ( y e. A -> ph ) )
9	8	alimi 1384	. . . . . 6 |- ( A. y y = x -> A. y ( y e. A -> ph ) )
10	9	a1d 22	. . . . 5 |- ( A. y y = x -> ( x e. A -> A. y ( y e. A -> ph ) ) )
11		nfr 1451	. . . . . 6 |- ( F/ y x e. A -> ( x e. A -> A. y x e. A ) )
12	6	alimi 1384	. . . . . 6 |- ( A. y x e. A -> A. y ( y e. A -> ph ) )
13	11, 12	syl6 33	. . . . 5 |- ( F/ y x e. A -> ( x e. A -> A. y ( y e. A -> ph ) ) )
14	10, 13	jaoi 668	. . . 4 |- ( ( A. y y = x \/ F/ y x e. A ) -> ( x e. A -> A. y ( y e. A -> ph ) ) )
15	3, 14	ax-mp 7	. . 3 |- ( x e. A -> A. y ( y e. A -> ph ) )
16		df-ral 2353	. . 3 |- ( A. y e. A ph <-> A. y ( y e. A -> ph ) )
17	15, 16	sylibr 132	. 2 |- ( x e. A -> A. y e. A ph )
18	17	rgen 2416	1 |- A. x e. A A. y e. A ph

Syntax hints:   -> wi 4   /\ wa 102   \/ wo 661   A.wal 1282   = wceq 1284   F/wnf 1389   e. wcel 1433   A.wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-cleq 2074  df-clel 2077  df-ral 2353

This theorem is referenced by:  ordsucunielexmid  4274  onintexmid  4315  isoid  5470  issmo  5926  ecopover  6227  ecopoverg  6230  subf  7310  negiso  8033  cnref1o  8733  ioof  8994  fzof  9154  gcdf  10364  eucalgf  10437  qredeu  10479