Theorem prvlem1

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

	Ref	Expression
Hypothesis	prvlem1.1	⊢ ( 𝜑 → 𝜓 )
Assertion	prvlem1	⊢ ( Prv 𝜑 → Prv 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		prvlem1.1	⊢ ( 𝜑 → 𝜓 )
2	1	ax-prv1	⊢ Prv ( 𝜑 → 𝜓 )
3		ax-prv2	⊢ ( Prv ( 𝜑 → 𝜓 ) → ( Prv 𝜑 → Prv 𝜓 ) )
4	2 3	ax-mp	⊢ ( Prv 𝜑 → Prv 𝜓 )