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Theorem sbctt 3769
 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 3701 . . . . 5 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
21bibi1d 347 . . . 4 (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑𝜑) ↔ ([𝐴 / 𝑥]𝜑𝜑)))
32imbi2d 344 . . 3 (𝑦 = 𝐴 → ((Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝜑)) ↔ (Ⅎ𝑥𝜑 → ([𝐴 / 𝑥]𝜑𝜑))))
4 sbft 2267 . . 3 (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝜑))
53, 4vtoclg 3487 . 2 (𝐴𝑉 → (Ⅎ𝑥𝜑 → ([𝐴 / 𝑥]𝜑𝜑)))
65imp 410 1 ((𝐴𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  Ⅎwnf 1785  [wsb 2069   ∈ wcel 2111  [wsbc 3698 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-sbc 3699 This theorem is referenced by:  sbcgf  3770  csbtt  3824
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