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Theorem rb-ax1 1517
 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
StepHypRef Expression
1 orim2 814 . . 3 ((ψχ) → ((φ ψ) → (φ χ)))
2 imor 401 . . 3 ((ψχ) ↔ (¬ ψ χ))
3 imor 401 . . 3 (((φ ψ) → (φ χ)) ↔ (¬ (φ ψ) (φ χ)))
41, 2, 33imtr3i 256 . 2 ((¬ ψ χ) → (¬ (φ ψ) (φ χ)))
54imori 402 1 (¬ (¬ ψ χ) (¬ (φ ψ) (φ χ)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360 This theorem is referenced by:  rbsyl  1521  rblem1  1522  rblem2  1523  rblem4  1525  re2luk1  1530
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