Theorem csbie2t 3014

Description: Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3015). (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 1504	. 2 |- F/ x A. x A. y ( ( x = A /\ y = B ) -> C = D )
2		nfcvd 2256	. 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 1541	. . . 4 |- F/ y A. x A. y ( ( x = A /\ y = B ) -> C = D )
6		nfv 1491	. . . 4 |- F/ y x = A
7	5, 6	nfan 1527	. . 3 |- F/ y ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) /\ x = A )
8		nfcvd 2256	. . 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 1471	. . . . 5 |- ( A. y ( ( x = A /\ y = B ) -> C = D ) -> ( ( x = A /\ y = B ) -> C = D ) )
12	11	sps 1500	. . . 4 |- ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) -> ( ( x = A /\ y = B ) -> C = D ) )
13	12	impl 375	. . 3 |- ( ( ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) /\ x = A ) /\ y = B ) -> C = D )
14	7, 8, 10, 13	csbiedf 3006	. 2 |- ( ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) /\ x = A ) -> [_ B / y ]_ C = D )
15	1, 2, 4, 14	csbiedf 3006	1 |- ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) -> [_ A / x ]_ [_ B / y ]_ C = D )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1312   = wceq 1314   e. wcel 1463   _Vcvv 2657   [_csb 2971

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-sbc 2879  df-csb 2972

This theorem is referenced by:  csbie2  3015