Theorem rgen2a 2511

Description: Generalization rule for restricted quantification. Note that x and y are not required to be disjoint. This proof illustrates the use of dvelim 1997. Usage of rgen2 2543 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 1508	. . . . 5 |- F/ y z e. A
2		eleq1 2220	. . . . 5 |- ( z = x -> ( z e. A <-> x e. A ) )
3	1, 2	dvelimor 1998	. . . 4 |- ( A. y y = x \/ F/ y x e. A )
4		eleq1 2220	. . . . . . . . 9 |- ( y = x -> ( y e. A <-> x e. A ) )
5		rgen2a.1	. . . . . . . . . 10 |- ( ( x e. A /\ y e. A ) -> ph )
6	5	ex 114	. . . . . . . . 9 |- ( x e. A -> ( y e. A -> ph ) )
7	4, 6	syl6bi 162	. . . . . . . 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 1435	. . . . . 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 1498	. . . . . 6 |- ( F/ y x e. A -> ( x e. A -> A. y x e. A ) )
12	6	alimi 1435	. . . . . 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 706	. . . 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 2440	. . 3 |- ( A. y e. A ph <-> A. y ( y e. A -> ph ) )
17	15, 16	sylibr 133	. 2 |- ( x e. A -> A. y e. A ph )
18	17	rgen 2510	1 |- A. x e. A A. y e. A ph

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 698   A.wal 1333   = wceq 1335   F/wnf 1440   e. wcel 2128   A.wral 2435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-cleq 2150  df-clel 2153  df-ral 2440

This theorem is referenced by:  ordsucunielexmid  4492  onintexmid  4534  isoid  5762  issmo  6237  oawordriexmid  6419  ecopover  6580  ecopoverg  6583  1domsn  6766  unfiexmid  6864  axaddf  7790  axmulf  7791  subf  8081  negiso  8831  cnref1o  9565  xaddf  9754  ioof  9881  fzof  10052  xrnegiso  11170  reeff1  11608  gcdf  11871  eucalgf  11947  qredeu  11989  qnnen  12230  strsetsid  12293  hmeofn  12772  ismeti  12816  qtopbasss  12991  tgqioo  13017  peano4nninf  13649