Theorem csbie2 3098

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)

Hypotheses

Ref	Expression
csbie2t.1	|- A e. _V
csbie2t.2	|- B e. _V
csbie2.3	|- ( ( x = A /\ y = B ) -> C = D )

Assertion

Ref	Expression
csbie2	|- [_ 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 csbie2

Step	Hyp	Ref	Expression
1		csbie2.3	. . 3 |- ( ( x = A /\ y = B ) -> C = D )
2	1	gen2 1443	. 2 |- A. x A. y ( ( x = A /\ y = B ) -> C = D )
3		csbie2t.1	. . 3 |- A e. _V
4		csbie2t.2	. . 3 |- B e. _V
5	3, 4	csbie2t 3097	. 2 |- ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) -> [_ A / x ]_ [_ B / y ]_ C = D )
6	2, 5	ax-mp 5	1 |- [_ A / x ]_ [_ B / y ]_ C = D

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1346   = wceq 1348   e. wcel 2141   _Vcvv 2730   [_csb 3049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050

This theorem is referenced by:  fsumcnv  11400  fprodcnv  11588