Theorem merco1lem13 1729

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1713. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merco1lem13	⊢ ((((𝜑 → 𝜓) → (𝜒 → 𝜓)) → 𝜏) → (𝜑 → 𝜏))

Proof of Theorem merco1lem13

Step	Hyp	Ref	Expression
1		merco1 1713	. . . 4 ⊢ (((((𝜓 → 𝜑) → (𝜒 → ⊥)) → 𝜑) → 𝜑) → ((𝜑 → 𝜓) → (𝜒 → 𝜓)))
2		merco1lem4 1719	. . . 4 ⊢ ((((((𝜓 → 𝜑) → (𝜒 → ⊥)) → 𝜑) → 𝜑) → ((𝜑 → 𝜓) → (𝜒 → 𝜓))) → (𝜑 → ((𝜑 → 𝜓) → (𝜒 → 𝜓))))
3	1, 2	ax-mp 5	. . 3 ⊢ (𝜑 → ((𝜑 → 𝜓) → (𝜒 → 𝜓)))
4		merco1lem12 1728	. . 3 ⊢ ((𝜑 → ((𝜑 → 𝜓) → (𝜒 → 𝜓))) → ((((𝜏 → 𝜑) → (𝜑 → ⊥)) → 𝜑) → ((𝜑 → 𝜓) → (𝜒 → 𝜓))))
5	3, 4	ax-mp 5	. 2 ⊢ ((((𝜏 → 𝜑) → (𝜑 → ⊥)) → 𝜑) → ((𝜑 → 𝜓) → (𝜒 → 𝜓)))
6		merco1 1713	. 2 ⊢ (((((𝜏 → 𝜑) → (𝜑 → ⊥)) → 𝜑) → ((𝜑 → 𝜓) → (𝜒 → 𝜓))) → ((((𝜑 → 𝜓) → (𝜒 → 𝜓)) → 𝜏) → (𝜑 → 𝜏)))
7	5, 6	ax-mp 5	1 ⊢ ((((𝜑 → 𝜓) → (𝜒 → 𝜓)) → 𝜏) → (𝜑 → 𝜏))

Syntax hints:   → wi 4  ⊥wfal 1552

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 207  df-tru 1543  df-fal 1553

This theorem is referenced by:  merco1lem14  1730  merco1lem15  1731  retbwax1  1735