Theorem opelopabsb 4294

Description: The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)

Assertion

Ref	Expression
opelopabsb	|- ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph )

Distinct variable groups:   x, y   x, B

Allowed substitution hints:   ph( x, y)   A( x, y)   B( y)

Proof of Theorem opelopabsb

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

Step	Hyp	Ref	Expression
1		elopab 4292	. . . 4 |- ( <. A , B >. e. { <. u , v >. | [ u / x ] [ v / y ] ph } <-> E. u E. v ( <. A , B >. = <. u , v >. /\ [ u / x ] [ v / y ] ph ) )
2		simpl 109	. . . . . . . 8 |- ( ( <. A , B >. = <. u , v >. /\ [ u / x ] [ v / y ] ph ) -> <. A , B >. = <. u , v >. )
3	2	eqcomd 2202	. . . . . . 7 |- ( ( <. A , B >. = <. u , v >. /\ [ u / x ] [ v / y ] ph ) -> <. u , v >. = <. A , B >. )
4		vex 2766	. . . . . . . 8 |- u e. _V
5		vex 2766	. . . . . . . 8 |- v e. _V
6	4, 5	opth 4270	. . . . . . 7 |- ( <. u , v >. = <. A , B >. <-> ( u = A /\ v = B ) )
7	3, 6	sylib 122	. . . . . 6 |- ( ( <. A , B >. = <. u , v >. /\ [ u / x ] [ v / y ] ph ) -> ( u = A /\ v = B ) )
8	7	2eximi 1615	. . . . 5 |- ( E. u E. v ( <. A , B >. = <. u , v >. /\ [ u / x ] [ v / y ] ph ) -> E. u E. v ( u = A /\ v = B ) )
9		eeanv 1951	. . . . . 6 |- ( E. u E. v ( u = A /\ v = B ) <-> ( E. u u = A /\ E. v v = B ) )
10		isset 2769	. . . . . . 7 |- ( A e. _V <-> E. u u = A )
11		isset 2769	. . . . . . 7 |- ( B e. _V <-> E. v v = B )
12	10, 11	anbi12i 460	. . . . . 6 |- ( ( A e. _V /\ B e. _V ) <-> ( E. u u = A /\ E. v v = B ) )
13	9, 12	bitr4i 187	. . . . 5 |- ( E. u E. v ( u = A /\ v = B ) <-> ( A e. _V /\ B e. _V ) )
14	8, 13	sylib 122	. . . 4 |- ( E. u E. v ( <. A , B >. = <. u , v >. /\ [ u / x ] [ v / y ] ph ) -> ( A e. _V /\ B e. _V ) )
15	1, 14	sylbi 121	. . 3 |- ( <. A , B >. e. { <. u , v >. | [ u / x ] [ v / y ] ph } -> ( A e. _V /\ B e. _V ) )
16		nfv 1542	. . . 4 |- F/ u ph
17		nfv 1542	. . . 4 |- F/ v ph
18		nfs1v 1958	. . . 4 |- F/ x [ u / x ] [ v / y ] ph
19		nfs1v 1958	. . . . 5 |- F/ y [ v / y ] ph
20	19	nfsbxy 1961	. . . 4 |- F/ y [ u / x ] [ v / y ] ph
21		sbequ12 1785	. . . . 5 |- ( y = v -> ( ph <-> [ v / y ] ph ) )
22		sbequ12 1785	. . . . 5 |- ( x = u -> ( [ v / y ] ph <-> [ u / x ] [ v / y ] ph ) )
23	21, 22	sylan9bbr 463	. . . 4 |- ( ( x = u /\ y = v ) -> ( ph <-> [ u / x ] [ v / y ] ph ) )
24	16, 17, 18, 20, 23	cbvopab 4104	. . 3 |- { <. x , y >. | ph } = { <. u , v >. | [ u / x ] [ v / y ] ph }
25	15, 24	eleq2s 2291	. 2 |- ( <. A , B >. e. { <. x , y >. | ph } -> ( A e. _V /\ B e. _V ) )
26		sbcex 2998	. . 3 |- ( [. A / x ]. [. B / y ]. ph -> A e. _V )
27		spesbc 3075	. . . 4 |- ( [. A / x ]. [. B / y ]. ph -> E. x [. B / y ]. ph )
28		sbcex 2998	. . . . 5 |- ( [. B / y ]. ph -> B e. _V )
29	28	exlimiv 1612	. . . 4 |- ( E. x [. B / y ]. ph -> B e. _V )
30	27, 29	syl 14	. . 3 |- ( [. A / x ]. [. B / y ]. ph -> B e. _V )
31	26, 30	jca 306	. 2 |- ( [. A / x ]. [. B / y ]. ph -> ( A e. _V /\ B e. _V ) )
32		opeq1 3808	. . . . 5 |- ( z = A -> <. z , w >. = <. A , w >. )
33	32	eleq1d 2265	. . . 4 |- ( z = A -> ( <. z , w >. e. { <. x , y >. | ph } <-> <. A , w >. e. { <. x , y >. | ph } ) )
34		dfsbcq2 2992	. . . 4 |- ( z = A -> ( [ z / x ] [ w / y ] ph <-> [. A / x ]. [ w / y ] ph ) )
35	33, 34	bibi12d 235	. . 3 |- ( z = A -> ( ( <. z , w >. e. { <. x , y >. | ph } <-> [ z / x ] [ w / y ] ph ) <-> ( <. A , w >. e. { <. x , y >. | ph } <-> [. A / x ]. [ w / y ] ph ) ) )
36		opeq2 3809	. . . . 5 |- ( w = B -> <. A , w >. = <. A , B >. )
37	36	eleq1d 2265	. . . 4 |- ( w = B -> ( <. A , w >. e. { <. x , y >. | ph } <-> <. A , B >. e. { <. x , y >. | ph } ) )
38		dfsbcq2 2992	. . . . 5 |- ( w = B -> ( [ w / y ] ph <-> [. B / y ]. ph ) )
39	38	sbcbidv 3048	. . . 4 |- ( w = B -> ( [. A / x ]. [ w / y ] ph <-> [. A / x ]. [. B / y ]. ph ) )
40	37, 39	bibi12d 235	. . 3 |- ( w = B -> ( ( <. A , w >. e. { <. x , y >. | ph } <-> [. A / x ]. [ w / y ] ph ) <-> ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph ) ) )
41		nfopab1 4102	. . . . . 6 |- F/_ x { <. x , y >. | ph }
42	41	nfel2 2352	. . . . 5 |- F/ x <. z , w >. e. { <. x , y >. | ph }
43		nfs1v 1958	. . . . 5 |- F/ x [ z / x ] [ w / y ] ph
44	42, 43	nfbi 1603	. . . 4 |- F/ x ( <. z , w >. e. { <. x , y >. | ph } <-> [ z / x ] [ w / y ] ph )
45		opeq1 3808	. . . . . 6 |- ( x = z -> <. x , w >. = <. z , w >. )
46	45	eleq1d 2265	. . . . 5 |- ( x = z -> ( <. x , w >. e. { <. x , y >. | ph } <-> <. z , w >. e. { <. x , y >. | ph } ) )
47		sbequ12 1785	. . . . 5 |- ( x = z -> ( [ w / y ] ph <-> [ z / x ] [ w / y ] ph ) )
48	46, 47	bibi12d 235	. . . 4 |- ( x = z -> ( ( <. x , w >. e. { <. x , y >. | ph } <-> [ w / y ] ph ) <-> ( <. z , w >. e. { <. x , y >. | ph } <-> [ z / x ] [ w / y ] ph ) ) )
49		nfopab2 4103	. . . . . . 7 |- F/_ y { <. x , y >. | ph }
50	49	nfel2 2352	. . . . . 6 |- F/ y <. x , w >. e. { <. x , y >. | ph }
51		nfs1v 1958	. . . . . 6 |- F/ y [ w / y ] ph
52	50, 51	nfbi 1603	. . . . 5 |- F/ y ( <. x , w >. e. { <. x , y >. | ph } <-> [ w / y ] ph )
53		opeq2 3809	. . . . . . 7 |- ( y = w -> <. x , y >. = <. x , w >. )
54	53	eleq1d 2265	. . . . . 6 |- ( y = w -> ( <. x , y >. e. { <. x , y >. | ph } <-> <. x , w >. e. { <. x , y >. | ph } ) )
55		sbequ12 1785	. . . . . 6 |- ( y = w -> ( ph <-> [ w / y ] ph ) )
56	54, 55	bibi12d 235	. . . . 5 |- ( y = w -> ( ( <. x , y >. e. { <. x , y >. | ph } <-> ph ) <-> ( <. x , w >. e. { <. x , y >. | ph } <-> [ w / y ] ph ) ) )
57		opabid 4290	. . . . 5 |- ( <. x , y >. e. { <. x , y >. | ph } <-> ph )
58	52, 56, 57	chvar 1771	. . . 4 |- ( <. x , w >. e. { <. x , y >. | ph } <-> [ w / y ] ph )
59	44, 48, 58	chvar 1771	. . 3 |- ( <. z , w >. e. { <. x , y >. | ph } <-> [ z / x ] [ w / y ] ph )
60	35, 40, 59	vtocl2g 2828	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph ) )
61	25, 31, 60	pm5.21nii 705	1 |- ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1506   [wsb 1776   e. wcel 2167   _Vcvv 2763   [.wsbc 2989   <.cop 3625   {copab 4093

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-opab 4095

This theorem is referenced by:  brabsb  4295  opelopabgf  4304  opelopabaf  4308  opelopabf  4309  difopab  4799  isarep1  5344