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Theorem sbctt 2891
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 2829 . . . . 5 (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
21bibi1d 231 . . . 4 (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑𝜑) ↔ ([𝐴 / 𝑥]𝜑𝜑)))
32imbi2d 228 . . 3 (𝑦 = 𝐴 → ((Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝜑)) ↔ (Ⅎ𝑥𝜑 → ([𝐴 / 𝑥]𝜑𝜑))))
4 sbft 1771 . . 3 (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑𝜑))
53, 4vtoclg 2669 . 2 (𝐴𝑉 → (Ⅎ𝑥𝜑 → ([𝐴 / 𝑥]𝜑𝜑)))
65imp 122 1 ((𝐴𝑉 ∧ Ⅎ𝑥𝜑) → ([𝐴 / 𝑥]𝜑𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wnf 1390  wcel 1434  [wsb 1687  [wsbc 2826
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-sbc 2827
This theorem is referenced by:  sbcgf  2892  csbtt  2929
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