Theorem csbie2t 3181

Description: Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3182). (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
csbie2t.1	|- A e. _V
csbie2t.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

Allowed substitution hints:   C(x,y)

Proof of Theorem csbie2t

Step	Hyp	Ref	Expression
1		nfa1 1788	. 2 |- F/xA.xA.y((x = A /\ y = B) -> C = D)
2		nfcvd 2491	. 2 |- (A.xA.y((x = A /\ y = B) -> C = D) -> F/_xD)
3		csbie2t.1	. . 3 |- A e. _V
4	3	a1i 10	. 2 |- (A.xA.y((x = A /\ y = B) -> C = D) -> A e. _V)
5		nfa2 1855	. . . 4 |- F/yA.xA.y((x = A /\ y = B) -> C = D)
6		nfv 1619	. . . 4 |- F/y x = A
7	5, 6	nfan 1824	. . 3 |- F/y(A.xA.y((x = A /\ y = B) -> C = D) /\ x = A)
8		nfcvd 2491	. . 3 |- ((A.xA.y((x = A /\ y = B) -> C = D) /\ x = A) -> F/_yD)
9		csbie2t.2	. . . 4 |- B e. _V
10	9	a1i 10	. . 3 |- ((A.xA.y((x = A /\ y = B) -> C = D) /\ x = A) -> B e. _V)
11		sp 1747	. . . . 5 |- (A.y((x = A /\ y = B) -> C = D) -> ((x = A /\ y = B) -> C = D))
12	11	sps 1754	. . . 4 |- (A.xA.y((x = A /\ y = B) -> C = D) -> ((x = A /\ y = B) -> C = D))
13	12	impl 603	. . 3 |- (((A.xA.y((x = A /\ y = B) -> C = D) /\ x = A) /\ y = B) -> C = D)
14	7, 8, 10, 13	csbiedf 3174	. 2 |- ((A.xA.y((x = A /\ y = B) -> C = D) /\ x = A) -> [_B / y]_C = D)
15	1, 2, 4, 14	csbiedf 3174	1 |- (A.xA.y((x = A /\ y = B) -> C = D) -> [_A / x]_[_B / y]_C = D)

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710  _Vcvv 2860  [_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:  csbie2  3182