Theorem pm4.15 830

Description: Theorem *4.15 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 18-Nov-2012.)

Assertion

Ref	Expression
pm4.15	⊢ (((𝜑 ∧ 𝜓) → ¬ 𝜒) ↔ ((𝜓 ∧ 𝜒) → ¬ 𝜑))

Proof of Theorem pm4.15

Step	Hyp	Ref	Expression
1		con2b 360	. 2 ⊢ (((𝜓 ∧ 𝜒) → ¬ 𝜑) ↔ (𝜑 → ¬ (𝜓 ∧ 𝜒)))
2		nan 827	. 2 ⊢ ((𝜑 → ¬ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) → ¬ 𝜒))
3	1, 2	bitr2i 275	1 ⊢ (((𝜑 ∧ 𝜓) → ¬ 𝜒) ↔ ((𝜓 ∧ 𝜒) → ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396

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 206  df-an 397

This theorem is referenced by: (None)