Theorem csbtt 3061

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 3050	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		nfcr 2304	. . . 4 |- ( F/_ x B -> F/ x y e. B )
3		sbctt 3021	. . . 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 2290	. 2 |- ( ( A e. V /\ F/_ x B ) -> { y | [. A / x ]. y e. B } = B )
6	1, 5	eqtrid 2215	1 |- ( ( A e. V /\ F/_ x B ) -> [_ A / x ]_ B = B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   F/wnf 1453   e. wcel 2141   {cab 2156   F/_wnfc 2299   [.wsbc 2955   [_csb 3049

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050

This theorem is referenced by:  csbconstgf  3062  sbnfc2  3109