Theorem csbie2t 2578

Description: Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 2579).

Hypotheses

Ref	Expression
csbie2g.1	|- A e. _V
csbie2g.2	|- B e. _V

Assertion

Ref	Expression
csbie2t	|- (A.xA.y((x = A /\ y = B) -> C = D) -> [_A / x]_[_B / y]_C = D)

Distinct variable groups:   x,y,A   x,B,y   x,D,y

Proof of Theorem csbie2t

Step	Hyp	Ref	Expression
1		csbie2g.1	. . 3 |- A e. _V
2	1	isseti 2297	. 2 |- E.x x = A
3		csbie2g.2	. . 3 |- B e. _V
4	3	isseti 2297	. 2 |- E.y y = B
5		hba1 1350	. . . 4 |- (A.xA.y((x = A /\ y = B) -> C = D) -> A.xA.xA.y((x = A /\ y = B) -> C = D))
6		ax-17 1317	. . . . . 6 |- (z e. A -> A.x z e. A)
7	1, 6	hbcsb1 2568	. . . . 5 |- (z e. [_A / x]_[_B / y]_C -> A.x z e. [_A / x]_[_B / y]_C)
8		ax-17 1317	. . . . 5 |- (z e. D -> A.x z e. D)
9	7, 8	hbeq 1995	. . . 4 |- ([_A / x]_[_B / y]_C = D -> A.x[_A / x]_[_B / y]_C = D)
10		hba1 1350	. . . . . 6 |- (A.y((x = A /\ y = B) -> C = D) -> A.yA.y((x = A /\ y = B) -> C = D))
11		ax-17 1317	. . . . . . . . 9 |- (z e. A -> A.y z e. A)
12		ax-17 1317	. . . . . . . . . 10 |- (z e. B -> A.y z e. B)
13	3, 12	hbcsb1 2568	. . . . . . . . 9 |- (z e. [_B / y]_C -> A.y z e. [_B / y]_C)
14	11, 13	hbcsbg 2569	. . . . . . . 8 |- (A e. _V -> (z e. [_A / x]_[_B / y]_C -> A.y z e. [_A / x]_[_B / y]_C))
15	1, 14	ax-mp 7	. . . . . . 7 |- (z e. [_A / x]_[_B / y]_C -> A.y z e. [_A / x]_[_B / y]_C)
16		ax-17 1317	. . . . . . 7 |- (z e. D -> A.y z e. D)
17	15, 16	hbeq 1995	. . . . . 6 |- ([_A / x]_[_B / y]_C = D -> A.y[_A / x]_[_B / y]_C = D)
18		csbeq1a 2546	. . . . . . . . . 10 |- (y = B -> C = [_B / y]_C)
19		csbeq1a 2546	. . . . . . . . . 10 |- (x = A -> [_B / y]_C = [_A / x]_[_B / y]_C)
20	18, 19	sylan9eqr 1951	. . . . . . . . 9 |- ((x = A /\ y = B) -> C = [_A / x]_[_B / y]_C)
21		pm3.43 664	. . . . . . . . 9 |- ((((x = A /\ y = B) -> C = [_A / x]_[_B / y]_C) /\ ((x = A /\ y = B) -> C = D)) -> ((x = A /\ y = B) -> (C = [_A / x]_[_B / y]_C /\ C = D)))
22	20, 21	mpan 759	. . . . . . . 8 |- (((x = A /\ y = B) -> C = D) -> ((x = A /\ y = B) -> (C = [_A / x]_[_B / y]_C /\ C = D)))
23		eqtr2 1905	. . . . . . . 8 |- ((C = [_A / x]_[_B / y]_C /\ C = D) -> [_A / x]_[_B / y]_C = D)
24	22, 23	syl6 25	. . . . . . 7 |- (((x = A /\ y = B) -> C = D) -> ((x = A /\ y = B) -> [_A / x]_[_B / y]_C = D))
25	24	a4s 1330	. . . . . 6 |- (A.y((x = A /\ y = B) -> C = D) -> ((x = A /\ y = B) -> [_A / x]_[_B / y]_C = D))
26	10, 17, 25	19.23ad 1415	. . . . 5 |- (A.y((x = A /\ y = B) -> C = D) -> (E.y(x = A /\ y = B) -> [_A / x]_[_B / y]_C = D))
27	26	a4s 1330	. . . 4 |- (A.xA.y((x = A /\ y = B) -> C = D) -> (E.y(x = A /\ y = B) -> [_A / x]_[_B / y]_C = D))
28	5, 9, 27	19.23ad 1415	. . 3 |- (A.xA.y((x = A /\ y = B) -> C = D) -> (E.xE.y(x = A /\ y = B) -> [_A / x]_[_B / y]_C = D))
29		eeanv 1707	. . 3 |- (E.xE.y(x = A /\ y = B) <-> (E.x x = A /\ E.y y = B))
30	28, 29	syl5ibr 224	. 2 |- (A.xA.y((x = A /\ y = B) -> C = D) -> ((E.x x = A /\ E.y y = B) -> [_A / x]_[_B / y]_C = D))
31	2, 4, 30	mp2ani 764	1 |- (A.xA.y((x = A /\ y = B) -> C = D) -> [_A / x]_[_B / y]_C = D)

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292  [_csb 2540

This theorem is referenced by:  csbie2 2579

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454  df-csb 2541

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