Theorem rblem1 1682

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
rblem1.1	⊢ (¬ 𝜑 ∨ 𝜓)
rblem1.2	⊢ (¬ 𝜒 ∨ 𝜃)

Assertion

Ref	Expression
rblem1	⊢ (¬ (𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜃))

Proof of Theorem rblem1

Step	Hyp	Ref	Expression
1		rblem1.2	. . 3 ⊢ (¬ 𝜒 ∨ 𝜃)
2		rb-ax1 1677	. . 3 ⊢ (¬ (¬ 𝜒 ∨ 𝜃) ∨ (¬ (𝜓 ∨ 𝜒) ∨ (𝜓 ∨ 𝜃)))
3	1, 2	anmp 1676	. 2 ⊢ (¬ (𝜓 ∨ 𝜒) ∨ (𝜓 ∨ 𝜃))
4		rb-ax2 1678	. . 3 ⊢ (¬ (𝜒 ∨ 𝜓) ∨ (𝜓 ∨ 𝜒))
5		rblem1.1	. . . . 5 ⊢ (¬ 𝜑 ∨ 𝜓)
6		rb-ax1 1677	. . . . 5 ⊢ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (¬ (𝜒 ∨ 𝜑) ∨ (𝜒 ∨ 𝜓)))
7	5, 6	anmp 1676	. . . 4 ⊢ (¬ (𝜒 ∨ 𝜑) ∨ (𝜒 ∨ 𝜓))
8		rb-ax2 1678	. . . 4 ⊢ (¬ (𝜑 ∨ 𝜒) ∨ (𝜒 ∨ 𝜑))
9	7, 8	rbsyl 1681	. . 3 ⊢ (¬ (𝜑 ∨ 𝜒) ∨ (𝜒 ∨ 𝜓))
10	4, 9	rbsyl 1681	. 2 ⊢ (¬ (𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜒))
11	3, 10	rbsyl 1681	1 ⊢ (¬ (𝜑 ∨ 𝜒) ∨ (𝜓 ∨ 𝜃))

Syntax hints:  ¬ wn 3   ∨ 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:  rblem4  1685  rblem5  1686  re2luk1  1690  re2luk2  1691