Theorem ralxpf 4808

Description: Version of ralxp 4805 with bound-variable hypotheses. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
ralxpf.1	|- F/ y ph
ralxpf.2	|- F/ z ph
ralxpf.3	|- F/ x ps
ralxpf.4	|- ( x = <. y , z >. -> ( ph <-> ps ) )

Assertion

Ref	Expression
ralxpf	|- ( A. x e. ( A X. B ) ph <-> A. y e. A A. z e. B ps )

Distinct variable groups:   x, y, A   x, z, B, y

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

Proof of Theorem ralxpf

Dummy variables v u w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvralsv 2742	. 2 |- ( A. x e. ( A X. B ) ph <-> A. v e. ( A X. B ) [ v / x ] ph )
2		cbvralsv 2742	. . . 4 |- ( A. z e. B [ w / y ] ps <-> A. u e. B [ u / z ] [ w / y ] ps )
3	2	ralbii 2500	. . 3 |- ( A. w e. A A. z e. B [ w / y ] ps <-> A. w e. A A. u e. B [ u / z ] [ w / y ] ps )
4		nfv 1539	. . . 4 |- F/ w A. z e. B ps
5		nfcv 2336	. . . . 5 |- F/_ y B
6		nfs1v 1955	. . . . 5 |- F/ y [ w / y ] ps
7	5, 6	nfralxy 2532	. . . 4 |- F/ y A. z e. B [ w / y ] ps
8		sbequ12 1782	. . . . 5 |- ( y = w -> ( ps <-> [ w / y ] ps ) )
9	8	ralbidv 2494	. . . 4 |- ( y = w -> ( A. z e. B ps <-> A. z e. B [ w / y ] ps ) )
10	4, 7, 9	cbvral 2722	. . 3 |- ( A. y e. A A. z e. B ps <-> A. w e. A A. z e. B [ w / y ] ps )
11		vex 2763	. . . . . 6 |- w e. _V
12		vex 2763	. . . . . 6 |- u e. _V
13	11, 12	eqvinop 4272	. . . . 5 |- ( v = <. w , u >. <-> E. y E. z ( v = <. y , z >. /\ <. y , z >. = <. w , u >. ) )
14		ralxpf.1	. . . . . . . 8 |- F/ y ph
15	14	nfsb 1962	. . . . . . 7 |- F/ y [ v / x ] ph
16	6	nfsb 1962	. . . . . . 7 |- F/ y [ u / z ] [ w / y ] ps
17	15, 16	nfbi 1600	. . . . . 6 |- F/ y ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps )
18		ralxpf.2	. . . . . . . . 9 |- F/ z ph
19	18	nfsb 1962	. . . . . . . 8 |- F/ z [ v / x ] ph
20		nfs1v 1955	. . . . . . . 8 |- F/ z [ u / z ] [ w / y ] ps
21	19, 20	nfbi 1600	. . . . . . 7 |- F/ z ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps )
22		ralxpf.3	. . . . . . . . 9 |- F/ x ps
23		ralxpf.4	. . . . . . . . 9 |- ( x = <. y , z >. -> ( ph <-> ps ) )
24	22, 23	sbhypf 2809	. . . . . . . 8 |- ( v = <. y , z >. -> ( [ v / x ] ph <-> ps ) )
25		vex 2763	. . . . . . . . . 10 |- y e. _V
26		vex 2763	. . . . . . . . . 10 |- z e. _V
27	25, 26	opth 4266	. . . . . . . . 9 |- ( <. y , z >. = <. w , u >. <-> ( y = w /\ z = u ) )
28		sbequ12 1782	. . . . . . . . . 10 |- ( z = u -> ( [ w / y ] ps <-> [ u / z ] [ w / y ] ps ) )
29	8, 28	sylan9bb 462	. . . . . . . . 9 |- ( ( y = w /\ z = u ) -> ( ps <-> [ u / z ] [ w / y ] ps ) )
30	27, 29	sylbi 121	. . . . . . . 8 |- ( <. y , z >. = <. w , u >. -> ( ps <-> [ u / z ] [ w / y ] ps ) )
31	24, 30	sylan9bb 462	. . . . . . 7 |- ( ( v = <. y , z >. /\ <. y , z >. = <. w , u >. ) -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
32	21, 31	exlimi 1605	. . . . . 6 |- ( E. z ( v = <. y , z >. /\ <. y , z >. = <. w , u >. ) -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
33	17, 32	exlimi 1605	. . . . 5 |- ( E. y E. z ( v = <. y , z >. /\ <. y , z >. = <. w , u >. ) -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
34	13, 33	sylbi 121	. . . 4 |- ( v = <. w , u >. -> ( [ v / x ] ph <-> [ u / z ] [ w / y ] ps ) )
35	34	ralxp 4805	. . 3 |- ( A. v e. ( A X. B ) [ v / x ] ph <-> A. w e. A A. u e. B [ u / z ] [ w / y ] ps )
36	3, 10, 35	3bitr4ri 213	. 2 |- ( A. v e. ( A X. B ) [ v / x ] ph <-> A. y e. A A. z e. B ps )
37	1, 36	bitri 184	1 |- ( A. x e. ( A X. B ) ph <-> A. y e. A A. z e. B ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   F/wnf 1471   E.wex 1503   [wsb 1773   A.wral 2472   <.cop 3621   X. cxp 4657

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-iun 3914  df-opab 4091  df-xp 4665  df-rel 4666

This theorem is referenced by: (None)