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

Step	Hyp	Ref	Expression
1		sbim 2065	. . 3 ⊢ ([y / x](¬ φ → ψ) ↔ ([y / x] ¬ φ → [y / x]ψ))
2		sbn 2062	. . . 4 ⊢ ([y / x] ¬ φ ↔ ¬ [y / x]φ)
3	2	imbi1i 315	. . 3 ⊢ (([y / x] ¬ φ → [y / x]ψ) ↔ (¬ [y / x]φ → [y / x]ψ))
4	1, 3	bitri 240	. 2 ⊢ ([y / x](¬ φ → ψ) ↔ (¬ [y / x]φ → [y / x]ψ))
5		df-or 359	. . 3 ⊢ ((φ ∨ ψ) ↔ (¬ φ → ψ))
6	5	sbbii 1653	. 2 ⊢ ([y / x](φ ∨ ψ) ↔ [y / x](¬ φ → ψ))
7		df-or 359	. 2 ⊢ (([y / x]φ ∨ [y / x]ψ) ↔ (¬ [y / x]φ → [y / x]ψ))
8	4, 6, 7	3bitr4i 268	1 ⊢ ([y / x](φ ∨ ψ) ↔ ([y / x]φ ∨ [y / x]ψ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357  [wsb 1648

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

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