Theorem csbie2 3177

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 1498	. 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 3176	. 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 104   A.wal 1395   = wceq 1397   e. wcel 2202   _Vcvv 2802   [_csb 3127

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-sbc 3032  df-csb 3128

This theorem is referenced by:  fsumcnv  11997  fprodcnv  12185  dfrhm2  14167  vtxdgfval  16138