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Theorem merco1lem3 1720
 Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1715. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merco1lem3 (((𝜑𝜓) → (𝜒 → ⊥)) → (𝜒𝜑))

Proof of Theorem merco1lem3
StepHypRef Expression
1 merco1lem2 1719 . . 3 (((𝜑𝜑) → ⊥) → (((𝜑𝜑) → (𝜑 → ⊥)) → ⊥))
2 retbwax2 1718 . . . 4 ((((𝜑𝜑) → (𝜑 → ⊥)) → (𝜑𝜑)) → (𝜑 → (((𝜑𝜑) → (𝜑 → ⊥)) → (𝜑𝜑))))
3 merco1lem2 1719 . . . 4 (((((𝜑𝜑) → (𝜑 → ⊥)) → (𝜑𝜑)) → (𝜑 → (((𝜑𝜑) → (𝜑 → ⊥)) → (𝜑𝜑)))) → ((((𝜑𝜑) → ⊥) → (((𝜑𝜑) → (𝜑 → ⊥)) → ⊥)) → (𝜑 → (((𝜑𝜑) → (𝜑 → ⊥)) → (𝜑𝜑)))))
42, 3ax-mp 5 . . 3 ((((𝜑𝜑) → ⊥) → (((𝜑𝜑) → (𝜑 → ⊥)) → ⊥)) → (𝜑 → (((𝜑𝜑) → (𝜑 → ⊥)) → (𝜑𝜑))))
51, 4ax-mp 5 . 2 (𝜑 → (((𝜑𝜑) → (𝜑 → ⊥)) → (𝜑𝜑)))
6 merco1lem2 1719 . . 3 (((𝜒𝜑) → ⊥) → (((𝜑𝜓) → (𝜒 → ⊥)) → ⊥))
7 retbwax2 1718 . . . 4 ((((𝜑𝜓) → (𝜒 → ⊥)) → (𝜒𝜑)) → ((𝜑 → (((𝜑𝜑) → (𝜑 → ⊥)) → (𝜑𝜑))) → (((𝜑𝜓) → (𝜒 → ⊥)) → (𝜒𝜑))))
8 merco1lem2 1719 . . . 4 (((((𝜑𝜓) → (𝜒 → ⊥)) → (𝜒𝜑)) → ((𝜑 → (((𝜑𝜑) → (𝜑 → ⊥)) → (𝜑𝜑))) → (((𝜑𝜓) → (𝜒 → ⊥)) → (𝜒𝜑)))) → ((((𝜒𝜑) → ⊥) → (((𝜑𝜓) → (𝜒 → ⊥)) → ⊥)) → ((𝜑 → (((𝜑𝜑) → (𝜑 → ⊥)) → (𝜑𝜑))) → (((𝜑𝜓) → (𝜒 → ⊥)) → (𝜒𝜑)))))
97, 8ax-mp 5 . . 3 ((((𝜒𝜑) → ⊥) → (((𝜑𝜓) → (𝜒 → ⊥)) → ⊥)) → ((𝜑 → (((𝜑𝜑) → (𝜑 → ⊥)) → (𝜑𝜑))) → (((𝜑𝜓) → (𝜒 → ⊥)) → (𝜒𝜑))))
106, 9ax-mp 5 . 2 ((𝜑 → (((𝜑𝜑) → (𝜑 → ⊥)) → (𝜑𝜑))) → (((𝜑𝜓) → (𝜒 → ⊥)) → (𝜒𝜑)))
115, 10ax-mp 5 1 (((𝜑𝜓) → (𝜒 → ⊥)) → (𝜒𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ⊥wfal 1550 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 210  df-tru 1541  df-fal 1551 This theorem is referenced by:  merco1lem4  1721  merco1lem6  1723  merco1lem11  1729  merco1lem12  1730  merco1lem18  1736
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