Theorem rp-frege3g

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

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

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

Proof

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