Theorem csbie2t 3173

Description: Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3174). (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. x A. 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 1587	. 2 |- F/ x A. x A. y ( ( x = A /\ y = B ) -> C = D )
2		nfcvd 2373	. 2 |- ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) -> F/_ x D )
3		csbie2t.1	. . 3 |- A e. _V
4	3	a1i 9	. 2 |- ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) -> A e. _V )
5		nfa2 1625	. . . 4 |- F/ y A. x A. y ( ( x = A /\ y = B ) -> C = D )
6		nfv 1574	. . . 4 |- F/ y x = A
7	5, 6	nfan 1611	. . 3 |- F/ y ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) /\ x = A )
8		nfcvd 2373	. . 3 |- ( ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) /\ x = A ) -> F/_ y D )
9		csbie2t.2	. . . 4 |- B e. _V
10	9	a1i 9	. . 3 |- ( ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) /\ x = A ) -> B e. _V )
11		sp 1557	. . . . 5 |- ( A. y ( ( x = A /\ y = B ) -> C = D ) -> ( ( x = A /\ y = B ) -> C = D ) )
12	11	sps 1583	. . . 4 |- ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) -> ( ( x = A /\ y = B ) -> C = D ) )
13	12	impl 380	. . 3 |- ( ( ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) /\ x = A ) /\ y = B ) -> C = D )
14	7, 8, 10, 13	csbiedf 3165	. 2 |- ( ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) /\ x = A ) -> [_ B / y ]_ C = D )
15	1, 2, 4, 14	csbiedf 3165	1 |- ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) -> [_ A / x ]_ [_ B / y ]_ C = D )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1393   = wceq 1395   e. wcel 2200   _Vcvv 2799   [_csb 3124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029  df-csb 3125

This theorem is referenced by:  csbie2  3174