Theorem sbctt 2880

Description: Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.)

Assertion

Ref	Expression
sbctt	|- ( ( A e. V /\ F/ x ph ) -> ( [. A / x ]. ph <-> ph ) )

Proof of Theorem sbctt

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 2818	. . . . 5 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
2	1	bibi1d 231	. . . 4 |- ( y = A -> ( ( [ y / x ] ph <-> ph ) <-> ( [. A / x ]. ph <-> ph ) ) )
3	2	imbi2d 228	. . 3 |- ( y = A -> ( ( F/ x ph -> ( [ y / x ] ph <-> ph ) ) <-> ( F/ x ph -> ( [. A / x ]. ph <-> ph ) ) ) )
4		sbft 1769	. . 3 |- ( F/ x ph -> ( [ y / x ] ph <-> ph ) )
5	3, 4	vtoclg 2658	. 2 |- ( A e. V -> ( F/ x ph -> ( [. A / x ]. ph <-> ph ) ) )
6	5	imp 122	1 |- ( ( A e. V /\ F/ x ph ) -> ( [. A / x ]. ph <-> ph ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   F/wnf 1389   e. wcel 1433   [wsb 1685   [.wsbc 2815

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816

This theorem is referenced by:  sbcgf  2881  csbtt  2918