New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  sbcan GIF version

Theorem sbcan 3088
 Description: Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)
Assertion
Ref Expression
sbcan ([̣A / x]̣(φ ψ) ↔ ([̣A / xφ A / xψ))

Proof of Theorem sbcan
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 sbcex 3055 . 2 ([̣A / x]̣(φ ψ) → A V)
2 sbcex 3055 . . 3 ([̣A / xψA V)
32adantl 452 . 2 (([̣A / xφ A / xψ) → A V)
4 dfsbcq2 3049 . . 3 (y = A → ([y / x](φ ψ) ↔ [̣A / x]̣(φ ψ)))
5 dfsbcq2 3049 . . . 4 (y = A → ([y / x]φ ↔ [̣A / xφ))
6 dfsbcq2 3049 . . . 4 (y = A → ([y / x]ψ ↔ [̣A / xψ))
75, 6anbi12d 691 . . 3 (y = A → (([y / x]φ [y / x]ψ) ↔ ([̣A / xφ A / xψ)))
8 sban 2069 . . 3 ([y / x](φ ψ) ↔ ([y / x]φ [y / x]ψ))
94, 7, 8vtoclbg 2915 . 2 (A V → ([̣A / x]̣(φ ψ) ↔ ([̣A / xφ A / xψ)))
101, 3, 9pm5.21nii 342 1 ([̣A / x]̣(φ ψ) ↔ ([̣A / xφ A / xψ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642  [wsb 1648   ∈ wcel 1710  Vcvv 2859  [̣wsbc 3046 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  inopab  4862
 Copyright terms: Public domain W3C validator