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 → (¬ 𝜑 → 𝜓))

Proof of Theorem re2luk3

Step	Hyp	Ref	Expression
1		rb-imdf 1752	. . . 4 ⊢ ¬ (¬ (¬ (¬ 𝜑 → 𝜓) ∨ (¬ ¬ 𝜑 ∨ 𝜓)) ∨ ¬ (¬ (¬ ¬ 𝜑 ∨ 𝜓) ∨ (¬ 𝜑 → 𝜓)))
2	1	rblem7 1765	. . 3 ⊢ (¬ (¬ ¬ 𝜑 ∨ 𝜓) ∨ (¬ 𝜑 → 𝜓))
3		rb-ax4 1757	. . . . . 6 ⊢ (¬ (¬ 𝜑 ∨ ¬ 𝜑) ∨ ¬ 𝜑)
4		rb-ax3 1756	. . . . . 6 ⊢ (¬ ¬ 𝜑 ∨ (¬ 𝜑 ∨ ¬ 𝜑))
5	3, 4	rbsyl 1758	. . . . 5 ⊢ (¬ ¬ 𝜑 ∨ ¬ 𝜑)
6		rb-ax2 1755	. . . . 5 ⊢ (¬ (¬ ¬ 𝜑 ∨ ¬ 𝜑) ∨ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
7	5, 6	anmp 1753	. . . 4 ⊢ (¬ 𝜑 ∨ ¬ ¬ 𝜑)
8		rblem2 1760	. . . 4 ⊢ (¬ (¬ 𝜑 ∨ ¬ ¬ 𝜑) ∨ (¬ 𝜑 ∨ (¬ ¬ 𝜑 ∨ 𝜓)))
9	7, 8	anmp 1753	. . 3 ⊢ (¬ 𝜑 ∨ (¬ ¬ 𝜑 ∨ 𝜓))
10	2, 9	rbsyl 1758	. 2 ⊢ (¬ 𝜑 ∨ (¬ 𝜑 → 𝜓))
11		rb-imdf 1752	. . 3 ⊢ ¬ (¬ (¬ (𝜑 → (¬ 𝜑 → 𝜓)) ∨ (¬ 𝜑 ∨ (¬ 𝜑 → 𝜓))) ∨ ¬ (¬ (¬ 𝜑 ∨ (¬ 𝜑 → 𝜓)) ∨ (𝜑 → (¬ 𝜑 → 𝜓))))
12	11	rblem7 1765	. 2 ⊢ (¬ (¬ 𝜑 ∨ (¬ 𝜑 → 𝜓)) ∨ (𝜑 → (¬ 𝜑 → 𝜓)))
13	10, 12	anmp 1753	1 ⊢ (𝜑 → (¬ 𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 845

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 206  df-an 397  df-or 846

This theorem is referenced by: (None)