Theorem imor 401

Description: Implication in terms of disjunction. Theorem *4.6 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
imor	⊢ ((φ → ψ) ↔ (¬ φ ∨ ψ))

Proof of Theorem imor

Step	Hyp	Ref	Expression
1		notnot 282	. . 3 ⊢ (φ ↔ ¬ ¬ φ)
2	1	imbi1i 315	. 2 ⊢ ((φ → ψ) ↔ (¬ ¬ φ → ψ))
3		df-or 359	. 2 ⊢ ((¬ φ ∨ ψ) ↔ (¬ ¬ φ → ψ))
4	2, 3	bitr4i 243	1 ⊢ ((φ → ψ) ↔ (¬ φ ∨ ψ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ 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

This theorem is referenced by:  imori  402  imorri  403  pm4.62  408  pm4.52  477  pm4.78  565  rb-bijust  1514  rb-imdf  1515  rb-ax1  1517  nf4  1868  r19.30  2757  dfimak2  4299  nncaddccl  4420  nndisjeq  4430  preaddccan2lem1  4455  ltfintrilem1  4466  evenoddnnnul  4515  leconnnc  6219  addccan2nclem2  6265  nchoicelem16  6305