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Theorem merco1lem4 1721
 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
merco1lem4 (((𝜑𝜓) → 𝜒) → (𝜓𝜒))

Proof of Theorem merco1lem4
StepHypRef Expression
1 merco1lem3 1720 . . 3 ((((𝜓 → ⊥) → (𝜑 → ⊥)) → ((𝜒𝜑) → ⊥)) → ((𝜒𝜑) → (𝜓 → ⊥)))
2 merco1 1715 . . 3 (((((𝜓 → ⊥) → (𝜑 → ⊥)) → ((𝜒𝜑) → ⊥)) → ((𝜒𝜑) → (𝜓 → ⊥))) → ((((𝜒𝜑) → (𝜓 → ⊥)) → 𝜓) → (𝜑𝜓)))
31, 2ax-mp 5 . 2 ((((𝜒𝜑) → (𝜓 → ⊥)) → 𝜓) → (𝜑𝜓))
4 merco1 1715 . 2 (((((𝜒𝜑) → (𝜓 → ⊥)) → 𝜓) → (𝜑𝜓)) → (((𝜑𝜓) → 𝜒) → (𝜓𝜒)))
53, 4ax-mp 5 1 (((𝜑𝜓) → 𝜒) → (𝜓𝜒))
 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:  merco1lem5  1722  merco1lem11  1729  merco1lem13  1731  merco1lem17  1735  merco1lem18  1736
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