Theorem pm4.14 561

Description: Theorem *4.14 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.)

Assertion

Ref	Expression
pm4.14	⊢ (((φ ∧ ψ) → χ) ↔ ((φ ∧ ¬ χ) → ¬ ψ))

Proof of Theorem pm4.14

Step	Hyp	Ref	Expression
1		con34b 283	. . 3 ⊢ ((ψ → χ) ↔ (¬ χ → ¬ ψ))
2	1	imbi2i 303	. 2 ⊢ ((φ → (ψ → χ)) ↔ (φ → (¬ χ → ¬ ψ)))
3		impexp 433	. 2 ⊢ (((φ ∧ ψ) → χ) ↔ (φ → (ψ → χ)))
4		impexp 433	. 2 ⊢ (((φ ∧ ¬ χ) → ¬ ψ) ↔ (φ → (¬ χ → ¬ ψ)))
5	2, 3, 4	3bitr4i 268	1 ⊢ (((φ ∧ ψ) → χ) ↔ ((φ ∧ ¬ χ) → ¬ ψ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358

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-an 360

This theorem is referenced by:  pm3.37  562