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

Step	Hyp	Ref	Expression
1		orim2 814	. . 3 ⊢ ((ψ → χ) → ((φ ∨ ψ) → (φ ∨ χ)))
2		imor 401	. . 3 ⊢ ((ψ → χ) ↔ (¬ ψ ∨ χ))
3		imor 401	. . 3 ⊢ (((φ ∨ ψ) → (φ ∨ χ)) ↔ (¬ (φ ∨ ψ) ∨ (φ ∨ χ)))
4	1, 2, 3	3imtr3i 256	. 2 ⊢ ((¬ ψ ∨ χ) → (¬ (φ ∨ ψ) ∨ (φ ∨ χ)))
5	4	imori 402	1 ⊢ (¬ (¬ ψ ∨ χ) ∨ (¬ (φ ∨ ψ) ∨ (φ ∨ χ)))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357

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 177  df-or 359  df-an 360

This theorem is referenced by:  rbsyl  1521  rblem1  1522  rblem2  1523  rblem4  1525  re2luk1  1530