Theorem rbsyl 1751

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
rbsyl.1	⊢ (¬ 𝜓 ∨ 𝜒)
rbsyl.2	⊢ (𝜑 ∨ 𝜓)

Assertion

Ref	Expression
rbsyl	⊢ (𝜑 ∨ 𝜒)

Proof of Theorem rbsyl

Step	Hyp	Ref	Expression
1		rbsyl.2	. 2 ⊢ (𝜑 ∨ 𝜓)
2		rbsyl.1	. . 3 ⊢ (¬ 𝜓 ∨ 𝜒)
3		rb-ax1 1747	. . 3 ⊢ (¬ (¬ 𝜓 ∨ 𝜒) ∨ (¬ (𝜑 ∨ 𝜓) ∨ (𝜑 ∨ 𝜒)))
4	2, 3	anmp 1746	. 2 ⊢ (¬ (𝜑 ∨ 𝜓) ∨ (𝜑 ∨ 𝜒))
5	1, 4	anmp 1746	1 ⊢ (𝜑 ∨ 𝜒)

Syntax hints:  ¬ wn 3   ∨ wo 846

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 396  df-or 847

This theorem is referenced by:  rblem1  1752  rblem2  1753  rblem3  1754  rblem4  1755  rblem5  1756  rblem6  1757  re2luk1  1760  re2luk2  1761  re2luk3  1762