Theorem sbctt 2970

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 2907	. . . . 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 1820	. . 3 |- ( F/ x ph -> ( [ y / x ] ph <-> ph ) )
5	3, 4	vtoclg 2741	. 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 1331   F/wnf 1436   e. wcel 1480   [wsb 1735   [.wsbc 2904

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sbc 2905

This theorem is referenced by:  sbcgf  2971  csbtt  3009