Theorem csbie 2973

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by AV, 2-Dec-2019.)

Hypotheses

Ref	Expression
csbie.1	|- A e. _V
csbie.2	|- ( x = A -> B = C )

Assertion

Ref	Expression
csbie	|- [_ A / x ]_ B = C

Distinct variable groups:   x, A   x, C

Allowed substitution hint:   B( x)

Proof of Theorem csbie

Step	Hyp	Ref	Expression
1		csbie.1	. 2 |- A e. _V
2		nfcv 2228	. 2 |- F/_ x C
3		csbie.2	. 2 |- ( x = A -> B = C )
4	1, 2, 3	csbief 2972	1 |- [_ A / x ]_ B = C

Syntax hints:   -> wi 4   = wceq 1289   e. wcel 1438   _Vcvv 2619   [_csb 2933

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-sbc 2841  df-csb 2934

This theorem is referenced by:  fsumcnv  10827  fisum0diag2  10837