Theorem merco1lem7 1723

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
merco1lem7	⊢ (𝜑 → (((𝜓 → 𝜒) → 𝜓) → 𝜓))

Proof of Theorem merco1lem7

Step	Hyp	Ref	Expression
1		merco1lem5 1721	. . 3 ⊢ ((((𝜓 → ⊥) → (((𝜓 → 𝜒) → 𝜓) → ⊥)) → 𝜒) → (𝜓 → 𝜒))
2		merco1 1714	. . 3 ⊢ (((((𝜓 → ⊥) → (((𝜓 → 𝜒) → 𝜓) → ⊥)) → 𝜒) → (𝜓 → 𝜒)) → (((𝜓 → 𝜒) → 𝜓) → (((𝜓 → 𝜒) → 𝜓) → 𝜓)))
3	1, 2	ax-mp 5	. 2 ⊢ (((𝜓 → 𝜒) → 𝜓) → (((𝜓 → 𝜒) → 𝜓) → 𝜓))
4		merco1lem6 1722	. 2 ⊢ ((((𝜓 → 𝜒) → 𝜓) → (((𝜓 → 𝜒) → 𝜓) → 𝜓)) → (𝜑 → (((𝜓 → 𝜒) → 𝜓) → 𝜓)))
5	3, 4	ax-mp 5	1 ⊢ (𝜑 → (((𝜓 → 𝜒) → 𝜓) → 𝜓))

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:  retbwax3  1724  merco1lem17  1734