Theorem csbiebt 2942

Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 2946.) (Contributed by NM, 11-Nov-2005.)

Assertion

Ref	Expression
csbiebt	|- ( ( A e. V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )

Distinct variable group:   x, A

Allowed substitution hints:   B( x)   C( x)   V( x)

Proof of Theorem csbiebt

Step	Hyp	Ref	Expression
1		elex 2610	. 2 |- ( A e. V -> A e. _V )
2		spsbc 2826	. . . . 5 |- ( A e. _V -> ( A. x ( x = A -> B = C ) -> [. A / x ]. ( x = A -> B = C ) ) )
3	2	adantr 270	. . . 4 |- ( ( A e. _V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) -> [. A / x ]. ( x = A -> B = C ) ) )
4		simpl 107	. . . . 5 |- ( ( A e. _V /\ F/_ x C ) -> A e. _V )
5		biimt 239	. . . . . . 7 |- ( x = A -> ( B = C <-> ( x = A -> B = C ) ) )
6		csbeq1a 2916	. . . . . . . 8 |- ( x = A -> B = [_ A / x ]_ B )
7	6	eqeq1d 2089	. . . . . . 7 |- ( x = A -> ( B = C <-> [_ A / x ]_ B = C ) )
8	5, 7	bitr3d 188	. . . . . 6 |- ( x = A -> ( ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
9	8	adantl 271	. . . . 5 |- ( ( ( A e. _V /\ F/_ x C ) /\ x = A ) -> ( ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
10		nfv 1461	. . . . . 6 |- F/ x A e. _V
11		nfnfc1 2222	. . . . . 6 |- F/ x F/_ x C
12	10, 11	nfan 1497	. . . . 5 |- F/ x ( A e. _V /\ F/_ x C )
13		nfcsb1v 2938	. . . . . . 7 |- F/_ x [_ A / x ]_ B
14	13	a1i 9	. . . . . 6 |- ( ( A e. _V /\ F/_ x C ) -> F/_ x [_ A / x ]_ B )
15		simpr 108	. . . . . 6 |- ( ( A e. _V /\ F/_ x C ) -> F/_ x C )
16	14, 15	nfeqd 2233	. . . . 5 |- ( ( A e. _V /\ F/_ x C ) -> F/ x [_ A / x ]_ B = C )
17	4, 9, 12, 16	sbciedf 2849	. . . 4 |- ( ( A e. _V /\ F/_ x C ) -> ( [. A / x ]. ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
18	3, 17	sylibd 147	. . 3 |- ( ( A e. _V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) -> [_ A / x ]_ B = C ) )
19	13	a1i 9	. . . . . . . 8 |- ( F/_ x C -> F/_ x [_ A / x ]_ B )
20		id 19	. . . . . . . 8 |- ( F/_ x C -> F/_ x C )
21	19, 20	nfeqd 2233	. . . . . . 7 |- ( F/_ x C -> F/ x [_ A / x ]_ B = C )
22	11, 21	nfan1 1496	. . . . . 6 |- F/ x ( F/_ x C /\ [_ A / x ]_ B = C )
23	7	biimprcd 158	. . . . . . 7 |- ( [_ A / x ]_ B = C -> ( x = A -> B = C ) )
24	23	adantl 271	. . . . . 6 |- ( ( F/_ x C /\ [_ A / x ]_ B = C ) -> ( x = A -> B = C ) )
25	22, 24	alrimi 1455	. . . . 5 |- ( ( F/_ x C /\ [_ A / x ]_ B = C ) -> A. x ( x = A -> B = C ) )
26	25	ex 113	. . . 4 |- ( F/_ x C -> ( [_ A / x ]_ B = C -> A. x ( x = A -> B = C ) ) )
27	26	adantl 271	. . 3 |- ( ( A e. _V /\ F/_ x C ) -> ( [_ A / x ]_ B = C -> A. x ( x = A -> B = C ) ) )
28	18, 27	impbid 127	. 2 |- ( ( A e. _V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
29	1, 28	sylan 277	1 |- ( ( A e. V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1282   = wceq 1284   e. wcel 1433   F/_wnfc 2206   _Vcvv 2601   [.wsbc 2815   [_csb 2908

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816  df-csb 2909

This theorem is referenced by:  csbiedf  2943  csbieb  2944  csbiegf  2946