Theorem prvlem2 37394

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

Hypothesis

Ref	Expression
prvlem2.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
prvlem2	⊢ (Prv 𝜑 → (Prv 𝜓 → Prv 𝜒))

Proof of Theorem prvlem2

Step	Hyp	Ref	Expression
1		prvlem2.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	prvlem1 37393	. 2 ⊢ (Prv 𝜑 → Prv (𝜓 → 𝜒))
3		ax-prv2 37391	. 2 ⊢ (Prv (𝜓 → 𝜒) → (Prv 𝜓 → Prv 𝜒))
4	2, 3	syl 18	1 ⊢ (Prv 𝜑 → (Prv 𝜓 → Prv 𝜒))

This proof depends on syntax axioms:   → wi 4  Prv cprvb 37389

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-prv1 37390  ax-prv2 37391

This theorem is used by:  bj-babygodel  37395  bj-babylob  37396