Theorem csbie2t 3562

Description: Conversion of implicit substitution to explicit substitution into a class (closed form of csbie2 3563). (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 2028	. 2 |- F/ x A. x A. y ( ( x = A /\ y = B ) -> C = D )
2		nfcvd 2765	. 2 |- ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) -> F/_ x D )
3		csbie2t.1	. . 3 |- A e. _V
4	3	a1i 11	. 2 |- ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) -> A e. _V )
5		nfa2 2040	. . . 4 |- F/ y A. x A. y ( ( x = A /\ y = B ) -> C = D )
6		nfv 1843	. . . 4 |- F/ y x = A
7	5, 6	nfan 1828	. . 3 |- F/ y ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) /\ x = A )
8		nfcvd 2765	. . 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 11	. . 3 |- ( ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) /\ x = A ) -> B e. _V )
11		2sp 2056	. . . 4 |- ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) -> ( ( x = A /\ y = B ) -> C = D ) )
12	11	impl 650	. . 3 |- ( ( ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) /\ x = A ) /\ y = B ) -> C = D )
13	7, 8, 10, 12	csbiedf 3554	. 2 |- ( ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) /\ x = A ) -> [_ B / y ]_ C = D )
14	1, 2, 4, 13	csbiedf 3554	1 |- ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) -> [_ A / x ]_ [_ B / y ]_ C = D )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534

This theorem is referenced by:  csbie2  3563