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Theorem rb-ax1 1826
 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 (¬ (¬ 𝜓𝜒) ∨ (¬ (𝜑𝜓) ∨ (𝜑𝜒)))

Proof of Theorem rb-ax1
StepHypRef Expression
1 orim2 922 . . 3 ((𝜓𝜒) → ((𝜑𝜓) → (𝜑𝜒)))
2 imor 427 . . 3 ((𝜓𝜒) ↔ (¬ 𝜓𝜒))
3 imor 427 . . 3 (((𝜑𝜓) → (𝜑𝜒)) ↔ (¬ (𝜑𝜓) ∨ (𝜑𝜒)))
41, 2, 33imtr3i 280 . 2 ((¬ 𝜓𝜒) → (¬ (𝜑𝜓) ∨ (𝜑𝜒)))
54imori 428 1 (¬ (¬ 𝜓𝜒) ∨ (¬ (𝜑𝜓) ∨ (𝜑𝜒)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ 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 197  df-or 384  df-an 385 This theorem is referenced by:  rbsyl  1830  rblem1  1831  rblem2  1832  rblem4  1834  re2luk1  1839
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