Theorem csbiebt 3553

Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3557.) (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 3212	. 2 |- ( A e. V -> A e. _V )
2		spsbc 3448	. . . . 5 |- ( A e. _V -> ( A. x ( x = A -> B = C ) -> [. A / x ]. ( x = A -> B = C ) ) )
3	2	adantr 481	. . . 4 |- ( ( A e. _V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) -> [. A / x ]. ( x = A -> B = C ) ) )
4		simpl 473	. . . . 5 |- ( ( A e. _V /\ F/_ x C ) -> A e. _V )
5		biimt 350	. . . . . . 7 |- ( x = A -> ( B = C <-> ( x = A -> B = C ) ) )
6		csbeq1a 3542	. . . . . . . 8 |- ( x = A -> B = [_ A / x ]_ B )
7	6	eqeq1d 2624	. . . . . . 7 |- ( x = A -> ( B = C <-> [_ A / x ]_ B = C ) )
8	5, 7	bitr3d 270	. . . . . 6 |- ( x = A -> ( ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
9	8	adantl 482	. . . . 5 |- ( ( ( A e. _V /\ F/_ x C ) /\ x = A ) -> ( ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
10		nfv 1843	. . . . . 6 |- F/ x A e. _V
11		nfnfc1 2767	. . . . . 6 |- F/ x F/_ x C
12	10, 11	nfan 1828	. . . . 5 |- F/ x ( A e. _V /\ F/_ x C )
13		nfcsb1v 3549	. . . . . . 7 |- F/_ x [_ A / x ]_ B
14	13	a1i 11	. . . . . 6 |- ( ( A e. _V /\ F/_ x C ) -> F/_ x [_ A / x ]_ B )
15		simpr 477	. . . . . 6 |- ( ( A e. _V /\ F/_ x C ) -> F/_ x C )
16	14, 15	nfeqd 2772	. . . . 5 |- ( ( A e. _V /\ F/_ x C ) -> F/ x [_ A / x ]_ B = C )
17	4, 9, 12, 16	sbciedf 3471	. . . 4 |- ( ( A e. _V /\ F/_ x C ) -> ( [. A / x ]. ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
18	3, 17	sylibd 229	. . 3 |- ( ( A e. _V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) -> [_ A / x ]_ B = C ) )
19	13	a1i 11	. . . . . . . 8 |- ( F/_ x C -> F/_ x [_ A / x ]_ B )
20		id 22	. . . . . . . 8 |- ( F/_ x C -> F/_ x C )
21	19, 20	nfeqd 2772	. . . . . . 7 |- ( F/_ x C -> F/ x [_ A / x ]_ B = C )
22	11, 21	nfan1 2068	. . . . . 6 |- F/ x ( F/_ x C /\ [_ A / x ]_ B = C )
23	7	biimprcd 240	. . . . . . 7 |- ( [_ A / x ]_ B = C -> ( x = A -> B = C ) )
24	23	adantl 482	. . . . . 6 |- ( ( F/_ x C /\ [_ A / x ]_ B = C ) -> ( x = A -> B = C ) )
25	22, 24	alrimi 2082	. . . . 5 |- ( ( F/_ x C /\ [_ A / x ]_ B = C ) -> A. x ( x = A -> B = C ) )
26	25	ex 450	. . . 4 |- ( F/_ x C -> ( [_ A / x ]_ B = C -> A. x ( x = A -> B = C ) ) )
27	26	adantl 482	. . 3 |- ( ( A e. _V /\ F/_ x C ) -> ( [_ A / x ]_ B = C -> A. x ( x = A -> B = C ) ) )
28	18, 27	impbid 202	. 2 |- ( ( A e. _V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )
29	1, 28	sylan 488	1 |- ( ( A e. V /\ F/_ x C ) -> ( A. x ( x = A -> B = C ) <-> [_ A / x ]_ B = C ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   F/_wnfc 2751   _Vcvv 3200   [.wsbc 3435   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534

This theorem is referenced by:  csbiedf  3554  csbieb  3555  csbiegf  3557