Theorem csbie2t 3146

Description: Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3147). (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 1565	. 2 |- F/ x A. x A. y ( ( x = A /\ y = B ) -> C = D )
2		nfcvd 2350	. 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 1603	. . . 4 |- F/ y A. x A. y ( ( x = A /\ y = B ) -> C = D )
6		nfv 1552	. . . 4 |- F/ y x = A
7	5, 6	nfan 1589	. . 3 |- F/ y ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) /\ x = A )
8		nfcvd 2350	. . 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 1535	. . . . 5 |- ( A. y ( ( x = A /\ y = B ) -> C = D ) -> ( ( x = A /\ y = B ) -> C = D ) )
12	11	sps 1561	. . . 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 3138	. 2 |- ( ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) /\ x = A ) -> [_ B / y ]_ C = D )
15	1, 2, 4, 14	csbiedf 3138	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 1371   = wceq 1373   e. wcel 2177   _Vcvv 2773   [_csb 3097

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-sbc 3003  df-csb 3098

This theorem is referenced by:  csbie2  3147