Theorem rbsyl 1672

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 1668	. . 3 ⊢ (¬ (¬ 𝜓 ∨ 𝜒) ∨ (¬ (𝜑 ∨ 𝜓) ∨ (𝜑 ∨ 𝜒)))
4	2, 3	anmp 1667	. 2 ⊢ (¬ (𝜑 ∨ 𝜓) ∨ (𝜑 ∨ 𝜒))
5	1, 4	anmp 1667	1 ⊢ (𝜑 ∨ 𝜒)

Syntax hints:  ¬ wn 3   ∨ wo 382

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 196  df-or 384  df-an 385

This theorem is referenced by:  rblem1  1673  rblem2  1674  rblem3  1675  rblem4  1676  rblem5  1677  rblem6  1678  re2luk1  1681  re2luk2  1682  re2luk3  1683