Theorem sbor2

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

	Ref	Expression
Assertion	sbor2	$$⊢\left(\left[t/x\right]\phi \vee \left[t/x\right]\psi \right)\to \left[t/x\right]\left(\phi \vee \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		orc	$$⊢\phi \to \left(\phi \vee \psi \right)$$
2	1	sbimi	$$⊢\left[t/x\right]\phi \to \left[t/x\right]\left(\phi \vee \psi \right)$$
3		olc	$$⊢\psi \to \left(\phi \vee \psi \right)$$
4	3	sbimi	$$⊢\left[t/x\right]\psi \to \left[t/x\right]\left(\phi \vee \psi \right)$$
5	2 4	jaoi	$$⊢\left(\left[t/x\right]\phi \vee \left[t/x\right]\psi \right)\to \left[t/x\right]\left(\phi \vee \psi \right)$$