Theorem sbor2 43009

Description: One direction of sbor 2341, using fewer axioms. Compare 19.33 1914. (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 880	. . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓))
2	1	sbimi 2108	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥](𝜑 ∨ 𝜓))
3		olc 881	. . 3 ⊢ (𝜓 → (𝜑 ∨ 𝜓))
4	3	sbimi 2108	. 2 ⊢ ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥](𝜑 ∨ 𝜓))
5	2, 4	jaoi 870	1 ⊢ (([𝑡 / 𝑥]𝜑 ∨ [𝑡 / 𝑥]𝜓) → [𝑡 / 𝑥](𝜑 ∨ 𝜓))

This proof depends on syntax axioms:   → wi 4   ∨ wo 860  [wsb 2096

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1810  df-sb 2097

This theorem is used by: (None)