Theorem sbctt 3052

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 2988	. . . . 5 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
2	1	bibi1d 233	. . . 4 |- ( y = A -> ( ( [ y / x ] ph <-> ph ) <-> ( [. A / x ]. ph <-> ph ) ) )
3	2	imbi2d 230	. . 3 |- ( y = A -> ( ( F/ x ph -> ( [ y / x ] ph <-> ph ) ) <-> ( F/ x ph -> ( [. A / x ]. ph <-> ph ) ) ) )
4		sbft 1859	. . 3 |- ( F/ x ph -> ( [ y / x ] ph <-> ph ) )
5	3, 4	vtoclg 2820	. 2 |- ( A e. V -> ( F/ x ph -> ( [. A / x ]. ph <-> ph ) ) )
6	5	imp 124	1 |- ( ( A e. V /\ F/ x ph ) -> ( [. A / x ]. ph <-> ph ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   F/wnf 1471   [wsb 1773   e. wcel 2164   [.wsbc 2985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-sbc 2986

This theorem is referenced by:  sbcgf  3053  csbtt  3092