Theorem csbtt 3057

Description: Substitution doesn't affect a constant B (in which x is not free). (Contributed by Mario Carneiro, 14-Oct-2016.)

Assertion

Ref	Expression
csbtt	|- ( ( A e. V /\ F/_ x B ) -> [_ A / x ]_ B = B )

Proof of Theorem csbtt

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3046	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		nfcr 2300	. . . 4 |- ( F/_ x B -> F/ x y e. B )
3		sbctt 3017	. . . 4 |- ( ( A e. V /\ F/ x y e. B ) -> ( [. A / x ]. y e. B <-> y e. B ) )
4	2, 3	sylan2 284	. . 3 |- ( ( A e. V /\ F/_ x B ) -> ( [. A / x ]. y e. B <-> y e. B ) )
5	4	abbi1dv 2286	. 2 |- ( ( A e. V /\ F/_ x B ) -> { y | [. A / x ]. y e. B } = B )
6	1, 5	syl5eq 2211	1 |- ( ( A e. V /\ F/_ x B ) -> [_ A / x ]_ B = B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   F/wnf 1448   e. wcel 2136   {cab 2151   F/_wnfc 2295   [.wsbc 2951   [_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-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:  csbconstgf  3058  sbnfc2  3105