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Theorem re1tbw4 1786
Description: tbw-ax4 1741 rederived from merco2 1774.

This theorem, along with re1tbw1 1783, re1tbw2 1784, and re1tbw3 1785, shows that merco2 1774, along with ax-mp 5, can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
re1tbw4 (⊥ → 𝜑)

Proof of Theorem re1tbw4
StepHypRef Expression
1 re1tbw3 1785 . . 3 (((𝜑𝜑) → 𝜑) → 𝜑)
2 re1tbw2 1784 . . . 4 (𝜑 → ((𝜑𝜑) → 𝜑))
3 re1tbw1 1783 . . . 4 ((𝜑 → ((𝜑𝜑) → 𝜑)) → ((((𝜑𝜑) → 𝜑) → 𝜑) → (𝜑𝜑)))
42, 3ax-mp 5 . . 3 ((((𝜑𝜑) → 𝜑) → 𝜑) → (𝜑𝜑))
51, 4ax-mp 5 . 2 (𝜑𝜑)
6 re1tbw3 1785 . . . . 5 ((((⊥ → 𝜑) → 𝜑) → (⊥ → 𝜑)) → (⊥ → 𝜑))
7 re1tbw2 1784 . . . . . 6 ((⊥ → 𝜑) → (((⊥ → 𝜑) → 𝜑) → (⊥ → 𝜑)))
8 re1tbw1 1783 . . . . . 6 (((⊥ → 𝜑) → (((⊥ → 𝜑) → 𝜑) → (⊥ → 𝜑))) → (((((⊥ → 𝜑) → 𝜑) → (⊥ → 𝜑)) → (⊥ → 𝜑)) → ((⊥ → 𝜑) → (⊥ → 𝜑))))
97, 8ax-mp 5 . . . . 5 (((((⊥ → 𝜑) → 𝜑) → (⊥ → 𝜑)) → (⊥ → 𝜑)) → ((⊥ → 𝜑) → (⊥ → 𝜑)))
106, 9ax-mp 5 . . . 4 ((⊥ → 𝜑) → (⊥ → 𝜑))
11 mercolem3 1777 . . . . 5 (((⊥ → 𝜑) → 𝜑) → ((⊥ → 𝜑) → (⊥ → 𝜑)))
12 merco2 1774 . . . . 5 ((((⊥ → 𝜑) → 𝜑) → ((⊥ → 𝜑) → (⊥ → 𝜑))) → (((⊥ → 𝜑) → (⊥ → 𝜑)) → ((𝜑𝜑) → ((𝜑𝜑) → (⊥ → 𝜑)))))
1311, 12ax-mp 5 . . . 4 (((⊥ → 𝜑) → (⊥ → 𝜑)) → ((𝜑𝜑) → ((𝜑𝜑) → (⊥ → 𝜑))))
1410, 13ax-mp 5 . . 3 ((𝜑𝜑) → ((𝜑𝜑) → (⊥ → 𝜑)))
155, 14ax-mp 5 . 2 ((𝜑𝜑) → (⊥ → 𝜑))
165, 15ax-mp 5 1 (⊥ → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wfal 1601
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 1599  df-fal 1602
This theorem is referenced by: (None)
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