Theorem rp-frege3g 45055

Description: Add antecedent to ax-frege2 45052. More general statement than frege3 45056. Like ax-frege2 45052, it is essentially a closed form of mpd 16, however it has an extra antecedent.

It would be more natural to prove from a1i 11 and ax-frege2 45052 in Metamath. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
rp-frege3g	⊢ (𝜑 → ((𝜓 → (𝜒 → 𝜃)) → ((𝜓 → 𝜒) → (𝜓 → 𝜃))))

Proof of Theorem rp-frege3g

Step	Hyp	Ref	Expression
1		ax-frege2 45052	. 2 ⊢ ((𝜓 → (𝜒 → 𝜃)) → ((𝜓 → 𝜒) → (𝜓 → 𝜃)))
2		ax-frege1 45051	. 2 ⊢ (((𝜓 → (𝜒 → 𝜃)) → ((𝜓 → 𝜒) → (𝜓 → 𝜃))) → (𝜑 → ((𝜓 → (𝜒 → 𝜃)) → ((𝜓 → 𝜒) → (𝜓 → 𝜃)))))
3	1, 2	ax-mp 5	1 ⊢ (𝜑 → ((𝜓 → (𝜒 → 𝜃)) → ((𝜓 → 𝜒) → (𝜓 → 𝜃))))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-frege1 45051  ax-frege2 45052

This theorem is used by:  rp-frege4g  45059