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Theorem sbctt 3003
 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 2940 . . . . 5 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
21bibi1d 232 . . . 4 (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑𝜑) ↔ ([𝐴 / 𝑥]𝜑𝜑)))
32imbi2d 229 . . 3 (𝑦 = 𝐴 → ((Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝜑)) ↔ (Ⅎ𝑥𝜑 → ([𝐴 / 𝑥]𝜑𝜑))))
4 sbft 1828 . . 3 (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝜑))
53, 4vtoclg 2772 . 2 (𝐴𝑉 → (Ⅎ𝑥𝜑 → ([𝐴 / 𝑥]𝜑𝜑)))
65imp 123 1 ((𝐴𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1335  Ⅎwnf 1440  [wsb 1742   ∈ wcel 2128  [wsbc 2937 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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-sbc 2938 This theorem is referenced by:  sbcgf  3004  csbtt  3043
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