Theorem csbie2 3094

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 1438	. 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 3093	. 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 1341   = wceq 1343   e. wcel 2136   _Vcvv 2726   [_csb 3045

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952  df-csb 3046

This theorem is referenced by:  fsumcnv  11378  fprodcnv  11566