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Theorem merco1lem4 1684
Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (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 1683 . . 3 ((((𝜓 → ⊥) → (𝜑 → ⊥)) → ((𝜒𝜑) → ⊥)) → ((𝜒𝜑) → (𝜓 → ⊥)))
2 merco1 1678 . . 3 (((((𝜓 → ⊥) → (𝜑 → ⊥)) → ((𝜒𝜑) → ⊥)) → ((𝜒𝜑) → (𝜓 → ⊥))) → ((((𝜒𝜑) → (𝜓 → ⊥)) → 𝜓) → (𝜑𝜓)))
31, 2ax-mp 5 . 2 ((((𝜒𝜑) → (𝜓 → ⊥)) → 𝜓) → (𝜑𝜓))
4 merco1 1678 . 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 1528
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 197  df-tru 1526  df-fal 1529
This theorem is referenced by:  merco1lem5  1685  merco1lem11  1692  merco1lem13  1694  merco1lem17  1698  merco1lem18  1699
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