Theorem csbiebt 3173

Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3177.) (Contributed by NM, 11-Nov-2005.)

Assertion

Ref	Expression
csbiebt	|- ((A e. V /\ F/_xC) -> (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 2868	. 2 |- (A e. V -> A e. _V)
2		spsbc 3059	. . . . 5 |- (A e. _V -> (A.x(x = A -> B = C) -> [.A / x ].(x = A -> B = C)))
3	2	adantr 451	. . . 4 |- ((A e. _V /\ F/_xC) -> (A.x(x = A -> B = C) -> [.A / x ].(x = A -> B = C)))
4		simpl 443	. . . . 5 |- ((A e. _V /\ F/_xC) -> A e. _V)
5		biimt 325	. . . . . . 7 |- (x = A -> (B = C <-> (x = A -> B = C)))
6		csbeq1a 3145	. . . . . . . 8 |- (x = A -> B = [_A / x]_B)
7	6	eqeq1d 2361	. . . . . . 7 |- (x = A -> (B = C <-> [_A / x]_B = C))
8	5, 7	bitr3d 246	. . . . . 6 |- (x = A -> ((x = A -> B = C) <-> [_A / x]_B = C))
9	8	adantl 452	. . . . 5 |- (((A e. _V /\ F/_xC) /\ x = A) -> ((x = A -> B = C) <-> [_A / x]_B = C))
10		nfv 1619	. . . . . 6 |- F/x A e. _V
11		nfnfc1 2493	. . . . . 6 |- F/x F/_xC
12	10, 11	nfan 1824	. . . . 5 |- F/x(A e. _V /\ F/_xC)
13		nfcsb1v 3169	. . . . . . 7 |- F/_x[_A / x]_B
14	13	a1i 10	. . . . . 6 |- ((A e. _V /\ F/_xC) -> F/_x[_A / x]_B)
15		simpr 447	. . . . . 6 |- ((A e. _V /\ F/_xC) -> F/_xC)
16	14, 15	nfeqd 2504	. . . . 5 |- ((A e. _V /\ F/_xC) -> F/x[_A / x]_B = C)
17	4, 9, 12, 16	sbciedf 3082	. . . 4 |- ((A e. _V /\ F/_xC) -> ( [.A / x ].(x = A -> B = C) <-> [_A / x]_B = C))
18	3, 17	sylibd 205	. . 3 |- ((A e. _V /\ F/_xC) -> (A.x(x = A -> B = C) -> [_A / x]_B = C))
19	13	a1i 10	. . . . . . . 8 |- ( F/_xC -> F/_x[_A / x]_B)
20		id 19	. . . . . . . 8 |- ( F/_xC -> F/_xC)
21	19, 20	nfeqd 2504	. . . . . . 7 |- ( F/_xC -> F/x[_A / x]_B = C)
22	11, 21	nfan1 1881	. . . . . 6 |- F/x( F/_xC /\ [_A / x]_B = C)
23	7	biimprcd 216	. . . . . . 7 |- ([_A / x]_B = C -> (x = A -> B = C))
24	23	adantl 452	. . . . . 6 |- (( F/_xC /\ [_A / x]_B = C) -> (x = A -> B = C))
25	22, 24	alrimi 1765	. . . . 5 |- (( F/_xC /\ [_A / x]_B = C) -> A.x(x = A -> B = C))
26	25	ex 423	. . . 4 |- ( F/_xC -> ([_A / x]_B = C -> A.x(x = A -> B = C)))
27	26	adantl 452	. . 3 |- ((A e. _V /\ F/_xC) -> ([_A / x]_B = C -> A.x(x = A -> B = C)))
28	18, 27	impbid 183	. 2 |- ((A e. _V /\ F/_xC) -> (A.x(x = A -> B = C) <-> [_A / x]_B = C))
29	1, 28	sylan 457	1 |- ((A e. V /\ F/_xC) -> (A.x(x = A -> B = C) <-> [_A / x]_B = C))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710   F/_wnfc 2477  _Vcvv 2860   [.wsbc 3047  [_csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by:  csbiedf  3174  csbieb  3175  csbiegf  3177