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Theorem sban 2069
 Description: Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sban ([y / x](φ ψ) ↔ ([y / x]φ [y / x]ψ))

Proof of Theorem sban
StepHypRef Expression
1 sbn 2062 . . 3 ([y / x] ¬ (φ → ¬ ψ) ↔ ¬ [y / x](φ → ¬ ψ))
2 sbim 2065 . . . 4 ([y / x](φ → ¬ ψ) ↔ ([y / x]φ → [y / x] ¬ ψ))
3 sbn 2062 . . . . 5 ([y / x] ¬ ψ ↔ ¬ [y / x]ψ)
43imbi2i 303 . . . 4 (([y / x]φ → [y / x] ¬ ψ) ↔ ([y / x]φ → ¬ [y / x]ψ))
52, 4bitri 240 . . 3 ([y / x](φ → ¬ ψ) ↔ ([y / x]φ → ¬ [y / x]ψ))
61, 5xchbinx 301 . 2 ([y / x] ¬ (φ → ¬ ψ) ↔ ¬ ([y / x]φ → ¬ [y / x]ψ))
7 df-an 360 . . 3 ((φ ψ) ↔ ¬ (φ → ¬ ψ))
87sbbii 1653 . 2 ([y / x](φ ψ) ↔ [y / x] ¬ (φ → ¬ ψ))
9 df-an 360 . 2 (([y / x]φ [y / x]ψ) ↔ ¬ ([y / x]φ → ¬ [y / x]ψ))
106, 8, 93bitr4i 268 1 ([y / x](φ ψ) ↔ ([y / x]φ [y / x]ψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  [wsb 1648 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  sb3an  2070  sbbi  2071  sbabel  2515  cbvreu  2833  sbcan  3088  sbcang  3089  rmo3  3133  inab  3522  difab  3523
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