Theorem pm3.37 690

Description: Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.)

Assertion

Ref	Expression
pm3.37	⊢ (((𝜑 ∧ 𝜓) → 𝜒) → ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓))

Proof of Theorem pm3.37

Step	Hyp	Ref	Expression
1		pm3.3 261	. . 3 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) → (𝜑 → (𝜓 → 𝜒)))
2		con3 643	. . 3 ⊢ ((𝜓 → 𝜒) → (¬ 𝜒 → ¬ 𝜓))
3	1, 2	syl6 33	. 2 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) → (𝜑 → (¬ 𝜒 → ¬ 𝜓)))
4	3	impd 254	1 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) → ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616

This theorem is referenced by:  ndvdssub  12095