Theorem sbor2

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

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

Proof

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