Theorem pm3.37 821

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		pm3.3 257	. . 3 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) → (𝜑 → (𝜓 → 𝜒)))
2		con3 603	. . 3 ⊢ ((𝜓 → 𝜒) → (¬ 𝜒 → ¬ 𝜓))
3	1, 2	syl6 33	. 2 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) → (𝜑 → (¬ 𝜒 → ¬ 𝜓)))
4	3	impd 251	1 ⊢ (((𝜑 ∧ 𝜓) → 𝜒) → ((𝜑 ∧ ¬ 𝜒) → ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577

This theorem is referenced by:  ndvdssub  10330