Theorem rb-ax4 1754

Description: The fourth of four axioms in the Russell-Bernays axiom system. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
rb-ax4	⊢ (¬ (𝜑 ∨ 𝜑) ∨ 𝜑)

Proof of Theorem rb-ax4

Step	Hyp	Ref	Expression
1		pm1.2 903	. . . 4 ⊢ ((𝜑 ∨ 𝜑) → 𝜑)
2	1	con3i 154	. . 3 ⊢ (¬ 𝜑 → ¬ (𝜑 ∨ 𝜑))
3	2	con1i 147	. 2 ⊢ (¬ ¬ (𝜑 ∨ 𝜑) → 𝜑)
4	3	orri 862	1 ⊢ (¬ (𝜑 ∨ 𝜑) ∨ 𝜑)

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 207  df-or 848

This theorem is referenced by:  rblem4  1759  rblem5  1760  rblem6  1761  re2luk1  1764  re2luk2  1765  re2luk3  1766