Theorem sbor 2398

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 2395	. . 3 ⊢ ([𝑦 / 𝑥](¬ 𝜑 → 𝜓) ↔ ([𝑦 / 𝑥] ¬ 𝜑 → [𝑦 / 𝑥]𝜓))
2		sbn 2391	. . . 4 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
3	2	imbi1i 339	. . 3 ⊢ (([𝑦 / 𝑥] ¬ 𝜑 → [𝑦 / 𝑥]𝜓) ↔ (¬ [𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
4	1, 3	bitri 264	. 2 ⊢ ([𝑦 / 𝑥](¬ 𝜑 → 𝜓) ↔ (¬ [𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
5		df-or 385	. . 3 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
6	5	sbbii 1887	. 2 ⊢ ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ [𝑦 / 𝑥](¬ 𝜑 → 𝜓))
7		df-or 385	. 2 ⊢ (([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓) ↔ (¬ [𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
8	4, 6, 7	3bitr4i 292	1 ⊢ ([𝑦 / 𝑥](𝜑 ∨ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∨ [𝑦 / 𝑥]𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383  [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbcor  3479  unab  3894  sbcorgOLD  38740