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Theorem merco1lem6 1686
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
merco1lem6 ((𝜑 → (𝜑𝜓)) → (𝜒 → (𝜑𝜓)))

Proof of Theorem merco1lem6
StepHypRef Expression
1 merco1lem5 1685 . . . . 5 (((((((𝜑𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → ⊥) → ⊥) → ((((𝜑𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥))
2 merco1lem3 1683 . . . . 5 ((((((((𝜑𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → ⊥) → ⊥) → ((((𝜑𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥)) → ((((𝜑𝜓) → ⊥) → (𝜒 → ⊥)) → (((((𝜑𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → ⊥)))
31, 2ax-mp 5 . . . 4 ((((𝜑𝜓) → ⊥) → (𝜒 → ⊥)) → (((((𝜑𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → ⊥))
4 merco1lem5 1685 . . . 4 (((((𝜑𝜓) → ⊥) → (𝜒 → ⊥)) → (((((𝜑𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → ⊥)) → ((𝜑𝜓) → (((((𝜑𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → ⊥)))
53, 4ax-mp 5 . . 3 ((𝜑𝜓) → (((((𝜑𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → ⊥))
6 merco1lem3 1683 . . 3 (((𝜑𝜓) → (((((𝜑𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → ⊥)) → (((((𝜑𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → 𝜑))
75, 6ax-mp 5 . 2 (((((𝜑𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → 𝜑)
8 merco1 1678 . 2 ((((((𝜑𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → 𝜑) → ((𝜑 → (𝜑𝜓)) → (𝜒 → (𝜑𝜓))))
97, 8ax-mp 5 1 ((𝜑 → (𝜑𝜓)) → (𝜒 → (𝜑𝜓)))
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:  merco1lem7  1687  merco1lem8  1689
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