Theorem csbtt 3071

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 3060	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		nfcr 2311	. . . 4 |- ( F/_ x B -> F/ x y e. B )
3		sbctt 3031	. . . 4 |- ( ( A e. V /\ F/ x y e. B ) -> ( [. A / x ]. y e. B <-> y e. B ) )
4	2, 3	sylan2 286	. . 3 |- ( ( A e. V /\ F/_ x B ) -> ( [. A / x ]. y e. B <-> y e. B ) )
5	4	abbi1dv 2297	. 2 |- ( ( A e. V /\ F/_ x B ) -> { y | [. A / x ]. y e. B } = B )
6	1, 5	eqtrid 2222	1 |- ( ( A e. V /\ F/_ x B ) -> [_ A / x ]_ B = B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   F/wnf 1460   e. wcel 2148   {cab 2163   F/_wnfc 2306   [.wsbc 2964   [_csb 3059

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-sbc 2965  df-csb 3060

This theorem is referenced by:  csbconstgf  3072  sbnfc2  3119