Theorem frege4 41407

Description: Special case of closed form of a2d 29. Special case of rp-frege4g 41406. Proposition 4 of [Frege1879] p. 31. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege4	⊢ (((𝜑 → 𝜓) → (𝜒 → (𝜑 → 𝜓))) → ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓))))

Proof of Theorem frege4

Step	Hyp	Ref	Expression
1		frege3 41403	. 2 ⊢ ((𝜑 → 𝜓) → ((𝜒 → (𝜑 → 𝜓)) → ((𝜒 → 𝜑) → (𝜒 → 𝜓))))
2		ax-frege2 41399	. 2 ⊢ (((𝜑 → 𝜓) → ((𝜒 → (𝜑 → 𝜓)) → ((𝜒 → 𝜑) → (𝜒 → 𝜓)))) → (((𝜑 → 𝜓) → (𝜒 → (𝜑 → 𝜓))) → ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓)))))
3	1, 2	ax-mp 5	1 ⊢ (((𝜑 → 𝜓) → (𝜒 → (𝜑 → 𝜓))) → ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓))))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-frege1 41398  ax-frege2 41399

This theorem is referenced by:  frege5  41408