Theorem rblem4 1768

Description: Used to rederive the Lukasiewicz axioms from Russell-Bernays'. (Contributed by Anthony Hart, 18-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
rblem4.1	⊢ (¬ 𝜑 ∨ 𝜃)
rblem4.2	⊢ (¬ 𝜓 ∨ 𝜏)
rblem4.3	⊢ (¬ 𝜒 ∨ 𝜂)

Assertion

Ref	Expression
rblem4	⊢ (¬ ((𝜑 ∨ 𝜓) ∨ 𝜒) ∨ ((𝜂 ∨ 𝜏) ∨ 𝜃))

Proof of Theorem rblem4

Step	Hyp	Ref	Expression
1		rblem4.3	. . . 4 ⊢ (¬ 𝜒 ∨ 𝜂)
2		rblem4.2	. . . 4 ⊢ (¬ 𝜓 ∨ 𝜏)
3	1, 2	rblem1 1765	. . 3 ⊢ (¬ (𝜒 ∨ 𝜓) ∨ (𝜂 ∨ 𝜏))
4		rblem4.1	. . 3 ⊢ (¬ 𝜑 ∨ 𝜃)
5	3, 4	rblem1 1765	. 2 ⊢ (¬ ((𝜒 ∨ 𝜓) ∨ 𝜑) ∨ ((𝜂 ∨ 𝜏) ∨ 𝜃))
6		rb-ax2 1761	. . . 4 ⊢ (¬ (𝜑 ∨ (𝜒 ∨ 𝜓)) ∨ ((𝜒 ∨ 𝜓) ∨ 𝜑))
7		rb-ax2 1761	. . . . . 6 ⊢ (¬ (𝜓 ∨ 𝜒) ∨ (𝜒 ∨ 𝜓))
8		rb-ax1 1760	. . . . . 6 ⊢ (¬ (¬ (𝜓 ∨ 𝜒) ∨ (𝜒 ∨ 𝜓)) ∨ (¬ (𝜑 ∨ (𝜓 ∨ 𝜒)) ∨ (𝜑 ∨ (𝜒 ∨ 𝜓))))
9	7, 8	anmp 1759	. . . . 5 ⊢ (¬ (𝜑 ∨ (𝜓 ∨ 𝜒)) ∨ (𝜑 ∨ (𝜒 ∨ 𝜓)))
10		rb-ax2 1761	. . . . 5 ⊢ (¬ ((𝜓 ∨ 𝜒) ∨ 𝜑) ∨ (𝜑 ∨ (𝜓 ∨ 𝜒)))
11	9, 10	rbsyl 1764	. . . 4 ⊢ (¬ ((𝜓 ∨ 𝜒) ∨ 𝜑) ∨ (𝜑 ∨ (𝜒 ∨ 𝜓)))
12	6, 11	rbsyl 1764	. . 3 ⊢ (¬ ((𝜓 ∨ 𝜒) ∨ 𝜑) ∨ ((𝜒 ∨ 𝜓) ∨ 𝜑))
13		rb-ax4 1763	. . . 4 ⊢ (¬ (((𝜓 ∨ 𝜒) ∨ 𝜑) ∨ ((𝜓 ∨ 𝜒) ∨ 𝜑)) ∨ ((𝜓 ∨ 𝜒) ∨ 𝜑))
14		rb-ax2 1761	. . . . . 6 ⊢ (¬ (𝜑 ∨ (𝜓 ∨ 𝜒)) ∨ ((𝜓 ∨ 𝜒) ∨ 𝜑))
15		rblem2 1766	. . . . . 6 ⊢ (¬ (𝜑 ∨ 𝜓) ∨ (𝜑 ∨ (𝜓 ∨ 𝜒)))
16	14, 15	rbsyl 1764	. . . . 5 ⊢ (¬ (𝜑 ∨ 𝜓) ∨ ((𝜓 ∨ 𝜒) ∨ 𝜑))
17		rb-ax3 1762	. . . . . 6 ⊢ (¬ 𝜒 ∨ (𝜓 ∨ 𝜒))
18		rblem2 1766	. . . . . 6 ⊢ (¬ (¬ 𝜒 ∨ (𝜓 ∨ 𝜒)) ∨ (¬ 𝜒 ∨ ((𝜓 ∨ 𝜒) ∨ 𝜑)))
19	17, 18	anmp 1759	. . . . 5 ⊢ (¬ 𝜒 ∨ ((𝜓 ∨ 𝜒) ∨ 𝜑))
20	16, 19	rblem1 1765	. . . 4 ⊢ (¬ ((𝜑 ∨ 𝜓) ∨ 𝜒) ∨ (((𝜓 ∨ 𝜒) ∨ 𝜑) ∨ ((𝜓 ∨ 𝜒) ∨ 𝜑)))
21	13, 20	rbsyl 1764	. . 3 ⊢ (¬ ((𝜑 ∨ 𝜓) ∨ 𝜒) ∨ ((𝜓 ∨ 𝜒) ∨ 𝜑))
22	12, 21	rbsyl 1764	. 2 ⊢ (¬ ((𝜑 ∨ 𝜓) ∨ 𝜒) ∨ ((𝜒 ∨ 𝜓) ∨ 𝜑))
23	5, 22	rbsyl 1764	1 ⊢ (¬ ((𝜑 ∨ 𝜓) ∨ 𝜒) ∨ ((𝜂 ∨ 𝜏) ∨ 𝜃))

Syntax hints:  ¬ wn 3   ∨ wo 847

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 848

This theorem is referenced by:  re2luk1  1773