Theorem sbctt 2979

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 2916	. . . . 5 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
2	1	bibi1d 232	. . . 4 |- ( y = A -> ( ( [ y / x ] ph <-> ph ) <-> ( [. A / x ]. ph <-> ph ) ) )
3	2	imbi2d 229	. . 3 |- ( y = A -> ( ( F/ x ph -> ( [ y / x ] ph <-> ph ) ) <-> ( F/ x ph -> ( [. A / x ]. ph <-> ph ) ) ) )
4		sbft 1821	. . 3 |- ( F/ x ph -> ( [ y / x ] ph <-> ph ) )
5	3, 4	vtoclg 2749	. 2 |- ( A e. V -> ( F/ x ph -> ( [. A / x ]. ph <-> ph ) ) )
6	5	imp 123	1 |- ( ( A e. V /\ F/ x ph ) -> ( [. A / x ]. ph <-> ph ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   F/wnf 1437   e. wcel 1481   [wsb 1736   [.wsbc 2913

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

This theorem is referenced by:  sbcgf  2980  csbtt  3019