Theorem cbvcsbv 3004

Description: Change the bound variable of a proper substitution into a class using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypothesis

Ref	Expression
cbvcsbv.1	|- ( x = y -> B = C )

Assertion

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

Distinct variable groups:   x, y   y, B   x, C

Allowed substitution hints:   A( x, y)   B( x)   C( y)

Proof of Theorem cbvcsbv

Step	Hyp	Ref	Expression
1		nfcv 2279	. 2 |- F/_ y B
2		nfcv 2279	. 2 |- F/_ x C
3		cbvcsbv.1	. 2 |- ( x = y -> B = C )
4	1, 2, 3	cbvcsb 3003	1 |- [_ A / x ]_ B = [_ A / y ]_ C

Syntax hints:   -> wi 4   = wceq 1331   [_csb 2998

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 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-sbc 2905  df-csb 2999

This theorem is referenced by: (None)