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Theorem sbctt 3778
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 ((𝐴𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑𝜑))

Proof of Theorem sbctt
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3714 . . . . 5 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
21bibi1d 345 . . . 4 (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑𝜑) ↔ ([𝐴 / 𝑥]𝜑𝜑)))
32imbi2d 342 . . 3 (𝑦 = 𝐴 → ((Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝜑)) ↔ (Ⅎ𝑥𝜑 → ([𝐴 / 𝑥]𝜑𝜑))))
4 sbft 2235 . . 3 (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝜑))
53, 4vtoclg 3513 . 2 (𝐴𝑉 → (Ⅎ𝑥𝜑 → ([𝐴 / 𝑥]𝜑𝜑)))
65imp 407 1 ((𝐴𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1525  wnf 1769  [wsb 2044  wcel 2083  [wsbc 3711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-12 2143  ax-ext 2771
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-sbc 3712
This theorem is referenced by:  sbcgf  3779  csbtt  3832
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