MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  re2luk3 Structured version   Visualization version   GIF version

Theorem re2luk3 1732
Description: luk-3 1622 derived from Russell-Bernays'.

This theorem, along with re1axmp 1729, re2luk1 1730, and re2luk2 1731 shows that rb-ax1 1717, rb-ax2 1718, rb-ax3 1719, and rb-ax4 1720, along with anmp 1716, 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 1715 . . . 4 ¬ (¬ (¬ (¬ 𝜑𝜓) ∨ (¬ ¬ 𝜑𝜓)) ∨ ¬ (¬ (¬ ¬ 𝜑𝜓) ∨ (¬ 𝜑𝜓)))
21rblem7 1728 . . 3 (¬ (¬ ¬ 𝜑𝜓) ∨ (¬ 𝜑𝜓))
3 rb-ax4 1720 . . . . . 6 (¬ (¬ 𝜑 ∨ ¬ 𝜑) ∨ ¬ 𝜑)
4 rb-ax3 1719 . . . . . 6 (¬ ¬ 𝜑 ∨ (¬ 𝜑 ∨ ¬ 𝜑))
53, 4rbsyl 1721 . . . . 5 (¬ ¬ 𝜑 ∨ ¬ 𝜑)
6 rb-ax2 1718 . . . . 5 (¬ (¬ ¬ 𝜑 ∨ ¬ 𝜑) ∨ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
75, 6anmp 1716 . . . 4 𝜑 ∨ ¬ ¬ 𝜑)
8 rblem2 1723 . . . 4 (¬ (¬ 𝜑 ∨ ¬ ¬ 𝜑) ∨ (¬ 𝜑 ∨ (¬ ¬ 𝜑𝜓)))
97, 8anmp 1716 . . 3 𝜑 ∨ (¬ ¬ 𝜑𝜓))
102, 9rbsyl 1721 . 2 𝜑 ∨ (¬ 𝜑𝜓))
11 rb-imdf 1715 . . 3 ¬ (¬ (¬ (𝜑 → (¬ 𝜑𝜓)) ∨ (¬ 𝜑 ∨ (¬ 𝜑𝜓))) ∨ ¬ (¬ (¬ 𝜑 ∨ (¬ 𝜑𝜓)) ∨ (𝜑 → (¬ 𝜑𝜓))))
1211rblem7 1728 . 2 (¬ (¬ 𝜑 ∨ (¬ 𝜑𝜓)) ∨ (𝜑 → (¬ 𝜑𝜓)))
1310, 12anmp 1716 1 (𝜑 → (¬ 𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382
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 384  df-an 385
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator