Theorem csbtt 3019

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 3008	. 2 |- [_ A / x ]_ B = { y | [. A / x ]. y e. B }
2		nfcr 2274	. . . 4 |- ( F/_ x B -> F/ x y e. B )
3		sbctt 2979	. . . 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 2260	. 2 |- ( ( A e. V /\ F/_ x B ) -> { y | [. A / x ]. y e. B } = B )
6	1, 5	syl5eq 2185	1 |- ( ( A e. V /\ F/_ x B ) -> [_ A / x ]_ B = B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   F/wnf 1437   e. wcel 1481   {cab 2126   F/_wnfc 2269   [.wsbc 2913   [_csb 3007

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sbc 2914  df-csb 3008

This theorem is referenced by:  csbconstgf  3020  sbnfc2  3065