Theorem rb-ax1 1751

Description: The first 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-ax1	⊢ (¬ (¬ 𝜓 ∨ 𝜒) ∨ (¬ (𝜑 ∨ 𝜓) ∨ (𝜑 ∨ 𝜒)))

Proof of Theorem rb-ax1

Step	Hyp	Ref	Expression
1		orim2 969	. . 3 ⊢ ((𝜓 → 𝜒) → ((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)))
2		imor 853	. . 3 ⊢ ((𝜓 → 𝜒) ↔ (¬ 𝜓 ∨ 𝜒))
3		imor 853	. . 3 ⊢ (((𝜑 ∨ 𝜓) → (𝜑 ∨ 𝜒)) ↔ (¬ (𝜑 ∨ 𝜓) ∨ (𝜑 ∨ 𝜒)))
4	1, 2, 3	3imtr3i 291	. 2 ⊢ ((¬ 𝜓 ∨ 𝜒) → (¬ (𝜑 ∨ 𝜓) ∨ (𝜑 ∨ 𝜒)))
5	4	imori 854	1 ⊢ (¬ (¬ 𝜓 ∨ 𝜒) ∨ (¬ (𝜑 ∨ 𝜓) ∨ (𝜑 ∨ 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ 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-an 396  df-or 848

This theorem is referenced by:  rbsyl  1755  rblem1  1756  rblem2  1757  rblem4  1759  re2luk1  1764