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Theorem sbctt 3108
 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 V xφ) → ([̣A / xφφ))

Proof of Theorem sbctt
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3049 . . . . 5 (y = A → ([y / x]φ ↔ [̣A / xφ))
21bibi1d 310 . . . 4 (y = A → (([y / x]φφ) ↔ ([̣A / xφφ)))
32imbi2d 307 . . 3 (y = A → ((Ⅎxφ → ([y / x]φφ)) ↔ (Ⅎxφ → ([̣A / xφφ))))
4 sbft 2025 . . 3 (Ⅎxφ → ([y / x]φφ))
53, 4vtoclg 2914 . 2 (A V → (Ⅎxφ → ([̣A / xφφ)))
65imp 418 1 ((A V xφ) → ([̣A / xφφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   = wceq 1642  [wsb 1648   ∈ wcel 1710  [̣wsbc 3046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047 This theorem is referenced by:  sbcgf  3109  csbtt  3148
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