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Theorem sbc3ang 3104
 Description: Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbc3ang (A V → ([̣A / x]̣(φ ψ χ) ↔ ([̣A / xφ A / xψ A / xχ)))

Proof of Theorem sbc3ang
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3049 . 2 (y = A → ([y / x](φ ψ χ) ↔ [̣A / x]̣(φ ψ χ)))
2 dfsbcq2 3049 . . 3 (y = A → ([y / x]φ ↔ [̣A / xφ))
3 dfsbcq2 3049 . . 3 (y = A → ([y / x]ψ ↔ [̣A / xψ))
4 dfsbcq2 3049 . . 3 (y = A → ([y / x]χ ↔ [̣A / xχ))
52, 3, 43anbi123d 1252 . 2 (y = A → (([y / x]φ [y / x]ψ [y / x]χ) ↔ ([̣A / xφ A / xψ A / xχ)))
6 sb3an 2070 . 2 ([y / x](φ ψ χ) ↔ ([y / x]φ [y / x]ψ [y / x]χ))
71, 5, 6vtoclbg 2915 1 (A V → ([̣A / x]̣(φ ψ χ) ↔ ([̣A / xφ A / xψ A / xχ)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934   = 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-3an 936  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: (None)
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