Theorem merco1lem4 1720

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1714. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merco1lem4	⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))

Proof of Theorem merco1lem4

Step	Hyp	Ref	Expression
1		merco1lem3 1719	. . 3 ⊢ ((((𝜓 → ⊥) → (𝜑 → ⊥)) → ((𝜒 → 𝜑) → ⊥)) → ((𝜒 → 𝜑) → (𝜓 → ⊥)))
2		merco1 1714	. . 3 ⊢ (((((𝜓 → ⊥) → (𝜑 → ⊥)) → ((𝜒 → 𝜑) → ⊥)) → ((𝜒 → 𝜑) → (𝜓 → ⊥))) → ((((𝜒 → 𝜑) → (𝜓 → ⊥)) → 𝜓) → (𝜑 → 𝜓)))
3	1, 2	ax-mp 5	. 2 ⊢ ((((𝜒 → 𝜑) → (𝜓 → ⊥)) → 𝜓) → (𝜑 → 𝜓))
4		merco1 1714	. 2 ⊢ (((((𝜒 → 𝜑) → (𝜓 → ⊥)) → 𝜓) → (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒)))
5	3, 4	ax-mp 5	1 ⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))

Syntax hints:   → wi 4  ⊥wfal 1549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-tru 1540  df-fal 1550

This theorem is referenced by:  merco1lem5  1721  merco1lem11  1728  merco1lem13  1730  merco1lem17  1734  merco1lem18  1735