Theorem csbie 3045

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 2281	. 2 |- F/_ x C
3		csbie.2	. 2 |- ( x = A -> B = C )
4	1, 2, 3	csbief 3044	1 |- [_ A / x ]_ B = C

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   _Vcvv 2686   [_csb 3003

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910  df-csb 3004

This theorem is referenced by:  fsumcnv  11209  fisum0diag2  11219