Theorem cbvcsbv 3130

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 2372	. 2 |- F/_ y B
2		nfcv 2372	. 2 |- F/_ x C
3		cbvcsbv.1	. 2 |- ( x = y -> B = C )
4	1, 2, 3	cbvcsb 3129	1 |- [_ A / x ]_ B = [_ A / y ]_ C

Syntax hints:   -> wi 4   = wceq 1395   [_csb 3124

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-sbc 3029  df-csb 3125

This theorem is referenced by: (None)