Theorem frege7 40152

Description: A closed form of syl6 35. The first antecedent is used to replace the consequent of the second antecedent. Proposition 7 of [Frege1879] p. 34. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege7	⊢ ((𝜑 → 𝜓) → ((𝜒 → (𝜃 → 𝜑)) → (𝜒 → (𝜃 → 𝜓))))

Proof of Theorem frege7

Step	Hyp	Ref	Expression
1		frege5 40144	. 2 ⊢ ((𝜑 → 𝜓) → ((𝜃 → 𝜑) → (𝜃 → 𝜓)))
2		frege6 40150	. 2 ⊢ (((𝜑 → 𝜓) → ((𝜃 → 𝜑) → (𝜃 → 𝜓))) → ((𝜑 → 𝜓) → ((𝜒 → (𝜃 → 𝜑)) → (𝜒 → (𝜃 → 𝜓)))))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → 𝜓) → ((𝜒 → (𝜃 → 𝜑)) → (𝜒 → (𝜃 → 𝜓))))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 40134  ax-frege2 40135

This theorem is referenced by:  frege32  40179  frege67a  40229  frege67b  40256  frege67c  40274  frege94  40301  frege107  40314  frege113  40320