Theorem csbtt 2862

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 2853	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		nfcr 2170	. . . 4 |- ( F/_ x B -> F/ x y e. B )
3		sbctt 2824	. . . 4 |- ( ( A e. V /\ F/ x y e. B ) -> ( [. A / x ]. y e. B <-> y e. B ) )
4	2, 3	sylan2 270	. . 3 |- ( ( A e. V /\ F/_ x B ) -> ( [. A / x ]. y e. B <-> y e. B ) )
5	4	abbi1dv 2157	. 2 |- ( ( A e. V /\ F/_ x B ) -> { y | [. A / x ]. y e. B } = B )
6	1, 5	syl5eq 2084	1 |- ( ( A e. V /\ F/_ x B ) -> [_ A / x ]_ B = B )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   F/wnf 1349   e. wcel 1393   {cab 2026   F/_wnfc 2165   [.wsbc 2764   [_csb 2852

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853

This theorem is referenced by:  csbconstgf  2863  sbnfc2  2906