Theorem merco1lem5 1753

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1746. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merco1lem5	⊢ ((((𝜑 → ⊥) → 𝜒) → 𝜏) → (𝜑 → 𝜏))

Proof of Theorem merco1lem5

Step	Hyp	Ref	Expression
1		merco1lem4 1752	. 2 ⊢ ((((𝜏 → 𝜑) → (𝜑 → ⊥)) → 𝜒) → ((𝜑 → ⊥) → 𝜒))
2		merco1 1746	. 2 ⊢ (((((𝜏 → 𝜑) → (𝜑 → ⊥)) → 𝜒) → ((𝜑 → ⊥) → 𝜒)) → ((((𝜑 → ⊥) → 𝜒) → 𝜏) → (𝜑 → 𝜏)))
3	1, 2	ax-mp 5	1 ⊢ ((((𝜑 → ⊥) → 𝜒) → 𝜏) → (𝜑 → 𝜏))

This proof depends on syntax axioms:   → wi 4  ⊥wfal 1582

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-tru 1573  df-fal 1583

This theorem is used by:  merco1lem6  1754  merco1lem7  1755  merco1lem11  1760  merco1lem18  1767