Theorem rgen2a 2587

Description: Generalization rule for restricted quantification. Note that x and y are not required to be disjoint. This proof illustrates the use of dvelim 2070. Usage of rgen2 2619 instead is highly encouraged. (Contributed by NM, 23-Nov-1994.) (Proof rewritten by Jim Kingdon, 1-Jun-2018.) (New usage is discouraged.)

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 1577	. . . . 5 |- F/ y z e. A
2		eleq1 2294	. . . . 5 |- ( z = x -> ( z e. A <-> x e. A ) )
3	1, 2	dvelimor 2071	. . . 4 |- ( A. y y = x \/ F/ y x e. A )
4		eleq1 2294	. . . . . . . . 9 |- ( y = x -> ( y e. A <-> x e. A ) )
5		rgen2a.1	. . . . . . . . . 10 |- ( ( x e. A /\ y e. A ) -> ph )
6	5	ex 115	. . . . . . . . 9 |- ( x e. A -> ( y e. A -> ph ) )
7	4, 6	biimtrdi 163	. . . . . . . 8 |- ( y = x -> ( y e. A -> ( y e. A -> ph ) ) )
8	7	pm2.43d 50	. . . . . . 7 |- ( y = x -> ( y e. A -> ph ) )
9	8	alimi 1504	. . . . . 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 1567	. . . . . 6 |- ( F/ y x e. A -> ( x e. A -> A. y x e. A ) )
12	6	alimi 1504	. . . . . 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 724	. . . 4 |- ( ( A. y y = x \/ F/ y x e. A ) -> ( x e. A -> A. y ( y e. A -> ph ) ) )
15	3, 14	ax-mp 5	. . 3 |- ( x e. A -> A. y ( y e. A -> ph ) )
16		df-ral 2516	. . 3 |- ( A. y e. A ph <-> A. y ( y e. A -> ph ) )
17	15, 16	sylibr 134	. 2 |- ( x e. A -> A. y e. A ph )
18	17	rgen 2586	1 |- A. x e. A A. y e. A ph

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   A.wal 1396   = wceq 1398   F/wnf 1509   e. wcel 2202   A.wral 2511

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-cleq 2224  df-clel 2227  df-ral 2516

This theorem is referenced by:  ordsucunielexmid  4635  onintexmid  4677  isoid  5961  issmo  6497  oawordriexmid  6681  ecopover  6845  ecopoverg  6848  1domsn  7044  unfiexmid  7153  axaddf  8148  axmulf  8149  subf  8440  negiso  9194  cnref1o  9946  xaddf  10140  ioof  10267  fzof  10441  xrnegiso  11902  reeff1  12341  gcdf  12623  eucalgf  12707  qredeu  12749  qnnen  13132  strsetsid  13195  hmeofn  15113  ismeti  15157  qtopbasss  15332  tgqioo  15366  peano4nninf  16732