Theorem sbor2 40105

Description: One direction of sbor 2307, using fewer axioms. Compare 19.33 1888. (Contributed by Steven Nguyen, 18-Aug-2023.)

Assertion

Ref	Expression
sbor2	⊢ (([𝑡 / 𝑥]𝜑 ∨ [𝑡 / 𝑥]𝜓) → [𝑡 / 𝑥](𝜑 ∨ 𝜓))

Proof of Theorem sbor2

Step	Hyp	Ref	Expression
1		orc 863	. . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
2	1	sbimi 2078	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥](𝜑 ∨ 𝜓))
3		olc 864	. . 3 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
4	3	sbimi 2078	. 2 ⊢ ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥](𝜑 ∨ 𝜓))
5	2, 4	jaoi 853	1 ⊢ (([𝑡 / 𝑥]𝜑 ∨ [𝑡 / 𝑥]𝜓) → [𝑡 / 𝑥](𝜑 ∨ 𝜓))

Syntax hints:   → wi 4   ∨ wo 843  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813

This theorem depends on definitions:  df-bi 206  df-or 844  df-sb 2069

This theorem is referenced by: (None)