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Theorem merco1lem7 1766
 Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1757. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merco1lem7 (𝜑 → (((𝜓𝜒) → 𝜓) → 𝜓))

Proof of Theorem merco1lem7
StepHypRef Expression
1 merco1lem5 1764 . . 3 ((((𝜓 → ⊥) → (((𝜓𝜒) → 𝜓) → ⊥)) → 𝜒) → (𝜓𝜒))
2 merco1 1757 . . 3 (((((𝜓 → ⊥) → (((𝜓𝜒) → 𝜓) → ⊥)) → 𝜒) → (𝜓𝜒)) → (((𝜓𝜒) → 𝜓) → (((𝜓𝜒) → 𝜓) → 𝜓)))
31, 2ax-mp 5 . 2 (((𝜓𝜒) → 𝜓) → (((𝜓𝜒) → 𝜓) → 𝜓))
4 merco1lem6 1765 . 2 ((((𝜓𝜒) → 𝜓) → (((𝜓𝜒) → 𝜓) → 𝜓)) → (𝜑 → (((𝜓𝜒) → 𝜓) → 𝜓)))
53, 4ax-mp 5 1 (𝜑 → (((𝜓𝜒) → 𝜓) → 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ⊥wfal 1614 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 199  df-tru 1605  df-fal 1615 This theorem is referenced by:  retbwax3  1767  merco1lem17  1777
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