Theorem csbiebt 3164

Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3168.) (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 2811	. 2 |- ( A e. V -> A e. _V )
2		spsbc 3040	. . . . 5 |- ( A e. _V -> ( A. x ( x = A -> B = C ) -> [. A / x ]. ( x = A -> B = C ) ) )
3	2	adantr 276	. . . 4 |- ( ( A e. _V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) -> [. A / x ]. ( x = A -> B = C ) ) )
4		simpl 109	. . . . 5 |- ( ( A e. _V /\ F/_ x C ) -> A e. _V )
5		biimt 241	. . . . . . 7 |- ( x = A -> ( B = C <-> ( x = A -> B = C ) ) )
6		csbeq1a 3133	. . . . . . . 8 |- ( x = A -> B = [_ A / x ]_ B )
7	6	eqeq1d 2238	. . . . . . 7 |- ( x = A -> ( B = C <-> [_ A / x ]_ B = C ) )
8	5, 7	bitr3d 190	. . . . . 6 |- ( x = A -> ( ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
9	8	adantl 277	. . . . 5 |- ( ( ( A e. _V /\ F/_ x C ) /\ x = A ) -> ( ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
10		nfv 1574	. . . . . 6 |- F/ x A e. _V
11		nfnfc1 2375	. . . . . 6 |- F/ x F/_ x C
12	10, 11	nfan 1611	. . . . 5 |- F/ x ( A e. _V /\ F/_ x C )
13		nfcsb1v 3157	. . . . . . 7 |- F/_ x [_ A / x ]_ B
14	13	a1i 9	. . . . . 6 |- ( ( A e. _V /\ F/_ x C ) -> F/_ x [_ A / x ]_ B )
15		simpr 110	. . . . . 6 |- ( ( A e. _V /\ F/_ x C ) -> F/_ x C )
16	14, 15	nfeqd 2387	. . . . 5 |- ( ( A e. _V /\ F/_ x C ) -> F/ x [_ A / x ]_ B = C )
17	4, 9, 12, 16	sbciedf 3064	. . . 4 |- ( ( A e. _V /\ F/_ x C ) -> ( [. A / x ]. ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
18	3, 17	sylibd 149	. . 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 2387	. . . . . . 7 |- ( F/_ x C -> F/ x [_ A / x ]_ B = C )
22	11, 21	nfan1 1610	. . . . . 6 |- F/ x ( F/_ x C /\ [_ A / x ]_ B = C )
23	7	biimprcd 160	. . . . . . 7 |- ( [_ A / x ]_ B = C -> ( x = A -> B = C ) )
24	23	adantl 277	. . . . . 6 |- ( ( F/_ x C /\ [_ A / x ]_ B = C ) -> ( x = A -> B = C ) )
25	22, 24	alrimi 1568	. . . . 5 |- ( ( F/_ x C /\ [_ A / x ]_ B = C ) -> A. x ( x = A -> B = C ) )
26	25	ex 115	. . . 4 |- ( F/_ x C -> ( [_ A / x ]_ B = C -> A. x ( x = A -> B = C ) ) )
27	26	adantl 277	. . 3 |- ( ( A e. _V /\ F/_ x C ) -> ( [_ A / x ]_ B = C -> A. x ( x = A -> B = C ) ) )
28	18, 27	impbid 129	. 2 |- ( ( A e. _V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
29	1, 28	sylan 283	1 |- ( ( A e. V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1393   = wceq 1395   e. wcel 2200   F/_wnfc 2359   _Vcvv 2799   [.wsbc 3028   [_csb 3124

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029  df-csb 3125

This theorem is referenced by:  csbiedf  3165  csbieb  3166  csbiegf  3168