Theorem csbiebt 3088

Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3092.) (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 2741	. 2 |- ( A e. V -> A e. _V )
2		spsbc 2966	. . . . 5 |- ( A e. _V -> ( A. x ( x = A -> B = C ) -> [. A / x ]. ( x = A -> B = C ) ) )
3	2	adantr 274	. . . 4 |- ( ( A e. _V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) -> [. A / x ]. ( x = A -> B = C ) ) )
4		simpl 108	. . . . 5 |- ( ( A e. _V /\ F/_ x C ) -> A e. _V )
5		biimt 240	. . . . . . 7 |- ( x = A -> ( B = C <-> ( x = A -> B = C ) ) )
6		csbeq1a 3058	. . . . . . . 8 |- ( x = A -> B = [_ A / x ]_ B )
7	6	eqeq1d 2179	. . . . . . 7 |- ( x = A -> ( B = C <-> [_ A / x ]_ B = C ) )
8	5, 7	bitr3d 189	. . . . . 6 |- ( x = A -> ( ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
9	8	adantl 275	. . . . 5 |- ( ( ( A e. _V /\ F/_ x C ) /\ x = A ) -> ( ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
10		nfv 1521	. . . . . 6 |- F/ x A e. _V
11		nfnfc1 2315	. . . . . 6 |- F/ x F/_ x C
12	10, 11	nfan 1558	. . . . 5 |- F/ x ( A e. _V /\ F/_ x C )
13		nfcsb1v 3082	. . . . . . 7 |- F/_ x [_ A / x ]_ B
14	13	a1i 9	. . . . . 6 |- ( ( A e. _V /\ F/_ x C ) -> F/_ x [_ A / x ]_ B )
15		simpr 109	. . . . . 6 |- ( ( A e. _V /\ F/_ x C ) -> F/_ x C )
16	14, 15	nfeqd 2327	. . . . 5 |- ( ( A e. _V /\ F/_ x C ) -> F/ x [_ A / x ]_ B = C )
17	4, 9, 12, 16	sbciedf 2990	. . . 4 |- ( ( A e. _V /\ F/_ x C ) -> ( [. A / x ]. ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
18	3, 17	sylibd 148	. . 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 2327	. . . . . . 7 |- ( F/_ x C -> F/ x [_ A / x ]_ B = C )
22	11, 21	nfan1 1557	. . . . . 6 |- F/ x ( F/_ x C /\ [_ A / x ]_ B = C )
23	7	biimprcd 159	. . . . . . 7 |- ( [_ A / x ]_ B = C -> ( x = A -> B = C ) )
24	23	adantl 275	. . . . . 6 |- ( ( F/_ x C /\ [_ A / x ]_ B = C ) -> ( x = A -> B = C ) )
25	22, 24	alrimi 1515	. . . . 5 |- ( ( F/_ x C /\ [_ A / x ]_ B = C ) -> A. x ( x = A -> B = C ) )
26	25	ex 114	. . . 4 |- ( F/_ x C -> ( [_ A / x ]_ B = C -> A. x ( x = A -> B = C ) ) )
27	26	adantl 275	. . 3 |- ( ( A e. _V /\ F/_ x C ) -> ( [_ A / x ]_ B = C -> A. x ( x = A -> B = C ) ) )
28	18, 27	impbid 128	. 2 |- ( ( A e. _V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
29	1, 28	sylan 281	1 |- ( ( A e. V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1346   = wceq 1348   e. wcel 2141   F/_wnfc 2299   _Vcvv 2730   [.wsbc 2955   [_csb 3049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050

This theorem is referenced by:  csbiedf  3089  csbieb  3090  csbiegf  3092