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Theorem re2luk2 1689
Description: luk-2 1579 derived from Russell-Bernays'. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
re2luk2 ((¬ 𝜑𝜑) → 𝜑)

Proof of Theorem re2luk2
StepHypRef Expression
1 rb-ax4 1678 . . . 4 (¬ (𝜑𝜑) ∨ 𝜑)
2 rb-ax3 1677 . . . . . . 7 𝜑 ∨ (𝜑𝜑))
31, 2rbsyl 1679 . . . . . 6 𝜑𝜑)
4 rb-ax4 1678 . . . . . . . . 9 (¬ (¬ ¬ 𝜑 ∨ ¬ ¬ 𝜑) ∨ ¬ ¬ 𝜑)
5 rb-ax3 1677 . . . . . . . . 9 (¬ ¬ ¬ 𝜑 ∨ (¬ ¬ 𝜑 ∨ ¬ ¬ 𝜑))
64, 5rbsyl 1679 . . . . . . . 8 (¬ ¬ ¬ 𝜑 ∨ ¬ ¬ 𝜑)
7 rb-ax2 1676 . . . . . . . 8 (¬ (¬ ¬ ¬ 𝜑 ∨ ¬ ¬ 𝜑) ∨ (¬ ¬ 𝜑 ∨ ¬ ¬ ¬ 𝜑))
86, 7anmp 1674 . . . . . . 7 (¬ ¬ 𝜑 ∨ ¬ ¬ ¬ 𝜑)
98, 3rblem1 1680 . . . . . 6 (¬ (¬ 𝜑𝜑) ∨ (¬ ¬ ¬ 𝜑𝜑))
103, 9anmp 1674 . . . . 5 (¬ ¬ ¬ 𝜑𝜑)
1110, 3rblem1 1680 . . . 4 (¬ (¬ ¬ 𝜑𝜑) ∨ (𝜑𝜑))
121, 11rbsyl 1679 . . 3 (¬ (¬ ¬ 𝜑𝜑) ∨ 𝜑)
13 rb-imdf 1673 . . . 4 ¬ (¬ (¬ (¬ 𝜑𝜑) ∨ (¬ ¬ 𝜑𝜑)) ∨ ¬ (¬ (¬ ¬ 𝜑𝜑) ∨ (¬ 𝜑𝜑)))
1413rblem6 1685 . . 3 (¬ (¬ 𝜑𝜑) ∨ (¬ ¬ 𝜑𝜑))
1512, 14rbsyl 1679 . 2 (¬ (¬ 𝜑𝜑) ∨ 𝜑)
16 rb-imdf 1673 . . 3 ¬ (¬ (¬ ((¬ 𝜑𝜑) → 𝜑) ∨ (¬ (¬ 𝜑𝜑) ∨ 𝜑)) ∨ ¬ (¬ (¬ (¬ 𝜑𝜑) ∨ 𝜑) ∨ ((¬ 𝜑𝜑) → 𝜑)))
1716rblem7 1686 . 2 (¬ (¬ (¬ 𝜑𝜑) ∨ 𝜑) ∨ ((¬ 𝜑𝜑) → 𝜑))
1815, 17anmp 1674 1 ((¬ 𝜑𝜑) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383
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-or 385  df-an 386
This theorem is referenced by: (None)
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