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Theorem sbor 2066
 Description: Logical OR inside and outside of substitution are equivalent. (Contributed by NM, 29-Sep-2002.)
Assertion
Ref Expression
sbor ([y / x](φ ψ) ↔ ([y / x]φ [y / x]ψ))

Proof of Theorem sbor
StepHypRef Expression
1 sbim 2065 . . 3 ([y / x](¬ φψ) ↔ ([y / x] ¬ φ → [y / x]ψ))
2 sbn 2062 . . . 4 ([y / x] ¬ φ ↔ ¬ [y / x]φ)
32imbi1i 315 . . 3 (([y / x] ¬ φ → [y / x]ψ) ↔ (¬ [y / x]φ → [y / x]ψ))
41, 3bitri 240 . 2 ([y / x](¬ φψ) ↔ (¬ [y / x]φ → [y / x]ψ))
5 df-or 359 . . 3 ((φ ψ) ↔ (¬ φψ))
65sbbii 1653 . 2 ([y / x](φ ψ) ↔ [y / x](¬ φψ))
7 df-or 359 . 2 (([y / x]φ [y / x]ψ) ↔ (¬ [y / x]φ → [y / x]ψ))
84, 6, 73bitr4i 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   ∨ wo 357  [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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649 This theorem is referenced by:  sbcor  3090  sbcorg  3091  unab  3521
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