Theorem annimim 775

Description: Express conjunction in terms of implication. One direction of Theorem *4.61 of [WhiteheadRussell] p. 120. The converse holds for decidable propositions, as can be seen at annimdc 833. (Contributed by Jim Kingdon, 24-Dec-2017.)

Assertion

Ref	Expression
annimim	⊢ ((φ ∧ ¬ ψ) → ¬ (φ → ψ))

Proof of Theorem annimim

Step	Hyp	Ref	Expression
1		pm2.27 35	. . 3 ⊢ (φ → ((φ → ψ) → ψ))
2		con3 558	. . 3 ⊢ (((φ → ψ) → ψ) → (¬ ψ → ¬ (φ → ψ)))
3	1, 2	syl 14	. 2 ⊢ (φ → (¬ ψ → ¬ (φ → ψ)))
4	3	imp 115	1 ⊢ ((φ ∧ ¬ ψ) → ¬ (φ → ψ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-in1 532  ax-in2 533

This theorem is referenced by:  pm4.65r  776  dcim  777  imanim  778  pm4.52im  796  exanaliim  1520