Theorem rgen2a 2584

Description: Generalization rule for restricted quantification. Note that x and y are not required to be disjoint. This proof illustrates the use of dvelim 2068. Usage of rgen2 2616 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 1574	. . . . 5 |- F/ y z e. A
2		eleq1 2292	. . . . 5 |- ( z = x -> ( z e. A <-> x e. A ) )
3	1, 2	dvelimor 2069	. . . 4 |- ( A. y y = x \/ F/ y x e. A )
4		eleq1 2292	. . . . . . . . 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 1501	. . . . . 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 1564	. . . . . 6 |- ( F/ y x e. A -> ( x e. A -> A. y x e. A ) )
12	6	alimi 1501	. . . . . 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 721	. . . 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 2513	. . 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 2583	1 |- A. x e. A A. y e. A ph

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   A.wal 1393   = wceq 1395   F/wnf 1506   e. wcel 2200   A.wral 2508

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-cleq 2222  df-clel 2225  df-ral 2513

This theorem is referenced by:  ordsucunielexmid  4623  onintexmid  4665  isoid  5934  issmo  6434  oawordriexmid  6616  ecopover  6780  ecopoverg  6783  1domsn  6976  unfiexmid  7080  axaddf  8055  axmulf  8056  subf  8348  negiso  9102  cnref1o  9846  xaddf  10040  ioof  10167  fzof  10340  xrnegiso  11773  reeff1  12211  gcdf  12493  eucalgf  12577  qredeu  12619  qnnen  13002  strsetsid  13065  hmeofn  14976  ismeti  15020  qtopbasss  15195  tgqioo  15229  peano4nninf  16372