Theorem merco1lem16 1735

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1716. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merco1lem16	⊢ (((𝜑 → (𝜓 → 𝜒)) → 𝜏) → ((𝜑 → 𝜒) → 𝜏))

Proof of Theorem merco1lem16

Step	Hyp	Ref	Expression
1		merco1lem15 1734	. . 3 ⊢ ((𝜑 → 𝜒) → (𝜑 → (𝜓 → 𝜒)))
2		merco1lem11 1730	. . 3 ⊢ (((𝜑 → 𝜒) → (𝜑 → (𝜓 → 𝜒))) → ((((𝜏 → 𝜑) → ((𝜑 → 𝜒) → ⊥)) → ⊥) → (𝜑 → (𝜓 → 𝜒))))
3	1, 2	ax-mp 5	. 2 ⊢ ((((𝜏 → 𝜑) → ((𝜑 → 𝜒) → ⊥)) → ⊥) → (𝜑 → (𝜓 → 𝜒)))
4		merco1 1716	. 2 ⊢ (((((𝜏 → 𝜑) → ((𝜑 → 𝜒) → ⊥)) → ⊥) → (𝜑 → (𝜓 → 𝜒))) → (((𝜑 → (𝜓 → 𝜒)) → 𝜏) → ((𝜑 → 𝜒) → 𝜏)))
5	3, 4	ax-mp 5	1 ⊢ (((𝜑 → (𝜓 → 𝜒)) → 𝜏) → ((𝜑 → 𝜒) → 𝜏))

Syntax hints:   → wi 4  ⊥wfal 1554

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 206  df-tru 1545  df-fal 1555

This theorem is referenced by:  merco1lem17  1736  retbwax1  1738