Theorem prvlem1

Description: An elementary property of the provability predicate. (Contributed by BJ, 3-Apr-2019)

	Ref	Expression
Hypothesis	prvlem1.1	$$⊢\phi \to \psi$$
Assertion	prvlem1	$$⊢Prv\phi \to Prv\psi$$

Proof

Step	Hyp	Ref	Expression
1		prvlem1.1	$$⊢\phi \to \psi$$
2	1	ax-prv1	$$⊢Prv\left(\phi \to \psi \right)$$
3		ax-prv2	$$⊢Prv\left(\phi \to \psi \right)\to \left(Prv\phi \to Prv\psi \right)$$
4	2 3	ax-mp	$$⊢Prv\phi \to Prv\psi$$