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Theorem re2luk3 1769
 Description: luk-3 1659 derived from Russell-Bernays'. This theorem, along with re1axmp 1766, re2luk1 1767, and re2luk2 1768 shows that rb-ax1 1754, rb-ax2 1755, rb-ax3 1756, and rb-ax4 1757, along with anmp 1753, can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 19-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
re2luk3 (𝜑 → (¬ 𝜑𝜓))

Proof of Theorem re2luk3
StepHypRef Expression
1 rb-imdf 1752 . . . 4 ¬ (¬ (¬ (¬ 𝜑𝜓) ∨ (¬ ¬ 𝜑𝜓)) ∨ ¬ (¬ (¬ ¬ 𝜑𝜓) ∨ (¬ 𝜑𝜓)))
21rblem7 1765 . . 3 (¬ (¬ ¬ 𝜑𝜓) ∨ (¬ 𝜑𝜓))
3 rb-ax4 1757 . . . . . 6 (¬ (¬ 𝜑 ∨ ¬ 𝜑) ∨ ¬ 𝜑)
4 rb-ax3 1756 . . . . . 6 (¬ ¬ 𝜑 ∨ (¬ 𝜑 ∨ ¬ 𝜑))
53, 4rbsyl 1758 . . . . 5 (¬ ¬ 𝜑 ∨ ¬ 𝜑)
6 rb-ax2 1755 . . . . 5 (¬ (¬ ¬ 𝜑 ∨ ¬ 𝜑) ∨ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
75, 6anmp 1753 . . . 4 𝜑 ∨ ¬ ¬ 𝜑)
8 rblem2 1760 . . . 4 (¬ (¬ 𝜑 ∨ ¬ ¬ 𝜑) ∨ (¬ 𝜑 ∨ (¬ ¬ 𝜑𝜓)))
97, 8anmp 1753 . . 3 𝜑 ∨ (¬ ¬ 𝜑𝜓))
102, 9rbsyl 1758 . 2 𝜑 ∨ (¬ 𝜑𝜓))
11 rb-imdf 1752 . . 3 ¬ (¬ (¬ (𝜑 → (¬ 𝜑𝜓)) ∨ (¬ 𝜑 ∨ (¬ 𝜑𝜓))) ∨ ¬ (¬ (¬ 𝜑 ∨ (¬ 𝜑𝜓)) ∨ (𝜑 → (¬ 𝜑𝜓))))
1211rblem7 1765 . 2 (¬ (¬ 𝜑 ∨ (¬ 𝜑𝜓)) ∨ (𝜑 → (¬ 𝜑𝜓)))
1310, 12anmp 1753 1 (𝜑 → (¬ 𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 844 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-an 400  df-or 845 This theorem is referenced by: (None)
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