Theorem merco1lem10 1729

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
merco1lem10	⊢ (((((𝜑 → 𝜓) → 𝜒) → (𝜏 → 𝜒)) → 𝜑) → (𝜃 → 𝜑))

Proof of Theorem merco1lem10

Step	Hyp	Ref	Expression
1		merco1 1716	. . 3 ⊢ (((((𝜒 → 𝜑) → (𝜏 → ⊥)) → 𝜑) → (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜒) → (𝜏 → 𝜒)))
2		merco1lem2 1720	. . 3 ⊢ ((((((𝜒 → 𝜑) → (𝜏 → ⊥)) → 𝜑) → (𝜑 → 𝜓)) → (((𝜑 → 𝜓) → 𝜒) → (𝜏 → 𝜒))) → ((((𝜑 → 𝜓) → (𝜃 → ⊥)) → ((((𝜒 → 𝜑) → (𝜏 → ⊥)) → 𝜑) → ⊥)) → (((𝜑 → 𝜓) → 𝜒) → (𝜏 → 𝜒))))
3	1, 2	ax-mp 5	. 2 ⊢ ((((𝜑 → 𝜓) → (𝜃 → ⊥)) → ((((𝜒 → 𝜑) → (𝜏 → ⊥)) → 𝜑) → ⊥)) → (((𝜑 → 𝜓) → 𝜒) → (𝜏 → 𝜒)))
4		merco1 1716	. 2 ⊢ (((((𝜑 → 𝜓) → (𝜃 → ⊥)) → ((((𝜒 → 𝜑) → (𝜏 → ⊥)) → 𝜑) → ⊥)) → (((𝜑 → 𝜓) → 𝜒) → (𝜏 → 𝜒))) → (((((𝜑 → 𝜓) → 𝜒) → (𝜏 → 𝜒)) → 𝜑) → (𝜃 → 𝜑)))
5	3, 4	ax-mp 5	1 ⊢ (((((𝜑 → 𝜓) → 𝜒) → (𝜏 → 𝜒)) → 𝜑) → (𝜃 → 𝜑))

Syntax hints:   → wi 4  ⊥wfal 1551

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 1542  df-fal 1552

This theorem is referenced by:  retbwax1  1738