Theorem rgen2a 2560

Description: Generalization rule for restricted quantification. Note that x and y are not required to be disjoint. This proof illustrates the use of dvelim 2045. Usage of rgen2 2592 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 1551	. . . . 5 |- F/ y z e. A
2		eleq1 2268	. . . . 5 |- ( z = x -> ( z e. A <-> x e. A ) )
3	1, 2	dvelimor 2046	. . . 4 |- ( A. y y = x \/ F/ y x e. A )
4		eleq1 2268	. . . . . . . . 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 1478	. . . . . 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 1541	. . . . . 6 |- ( F/ y x e. A -> ( x e. A -> A. y x e. A ) )
12	6	alimi 1478	. . . . . 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 718	. . . 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 2489	. . 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 2559	1 |- A. x e. A A. y e. A ph

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 710   A.wal 1371   = wceq 1373   F/wnf 1483   e. wcel 2176   A.wral 2484

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-cleq 2198  df-clel 2201  df-ral 2489

This theorem is referenced by:  ordsucunielexmid  4580  onintexmid  4622  isoid  5881  issmo  6376  oawordriexmid  6558  ecopover  6722  ecopoverg  6725  1domsn  6916  unfiexmid  7017  axaddf  7983  axmulf  7984  subf  8276  negiso  9030  cnref1o  9774  xaddf  9968  ioof  10095  fzof  10268  xrnegiso  11606  reeff1  12044  gcdf  12326  eucalgf  12410  qredeu  12452  qnnen  12835  strsetsid  12898  hmeofn  14807  ismeti  14851  qtopbasss  15026  tgqioo  15060  peano4nninf  15980