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Theorem rb-ax1 1760
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 968 . . 3 ((𝜓𝜒) → ((𝜑𝜓) → (𝜑𝜒)))
2 imor 853 . . 3 ((𝜓𝜒) ↔ (¬ 𝜓𝜒))
3 imor 853 . . 3 (((𝜑𝜓) → (𝜑𝜒)) ↔ (¬ (𝜑𝜓) ∨ (𝜑𝜒)))
41, 2, 33imtr3i 294 . 2 ((¬ 𝜓𝜒) → (¬ (𝜑𝜓) ∨ (𝜑𝜒)))
54imori 854 1 (¬ (¬ 𝜓𝜒) ∨ (¬ (𝜑𝜓) ∨ (𝜑𝜒)))
Colors of variables: wff setvar class
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 210  df-an 400  df-or 848
This theorem is referenced by:  rbsyl  1764  rblem1  1765  rblem2  1766  rblem4  1768  re2luk1  1773
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