Theorem rblem1 1522

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 1517	. . 3 ⊢ (¬ (¬ χ ∨ θ) ∨ (¬ (ψ ∨ χ) ∨ (ψ ∨ θ)))
3	1, 2	anmp 1516	. 2 ⊢ (¬ (ψ ∨ χ) ∨ (ψ ∨ θ))
4		rb-ax2 1518	. . 3 ⊢ (¬ (χ ∨ ψ) ∨ (ψ ∨ χ))
5		rblem1.1	. . . . 5 ⊢ (¬ φ ∨ ψ)
6		rb-ax1 1517	. . . . 5 ⊢ (¬ (¬ φ ∨ ψ) ∨ (¬ (χ ∨ φ) ∨ (χ ∨ ψ)))
7	5, 6	anmp 1516	. . . 4 ⊢ (¬ (χ ∨ φ) ∨ (χ ∨ ψ))
8		rb-ax2 1518	. . . 4 ⊢ (¬ (φ ∨ χ) ∨ (χ ∨ φ))
9	7, 8	rbsyl 1521	. . 3 ⊢ (¬ (φ ∨ χ) ∨ (χ ∨ ψ))
10	4, 9	rbsyl 1521	. 2 ⊢ (¬ (φ ∨ χ) ∨ (ψ ∨ χ))
11	3, 10	rbsyl 1521	1 ⊢ (¬ (φ ∨ χ) ∨ (ψ ∨ θ))

This proof depends on syntax axioms:  ¬ wn 3   ∨ wo 357

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360

This theorem is used by:  rblem4  1525  rblem5  1526  re2luk1  1530  re2luk2  1531