Theorem imasandint-P4.33b 490

Description: One Direction of '→' in Terms of '∧'. †

Only this direction is deducible with intuitionist logic.

Assertion

Ref	Expression
imasandint-P4.33b	⊢ ((𝜑 → 𝜓) → ¬ (𝜑 ∧ ¬ 𝜓))

Proof of Theorem imasandint-P4.33b

Step	Hyp	Ref	Expression
1		rcp-NDASM2of2 194	. . . 4 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 ∧ ¬ 𝜓)) → (𝜑 ∧ ¬ 𝜓))
2	1	ndander-P3.9 174	. . 3 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 ∧ ¬ 𝜓)) → 𝜑)
3		rcp-NDASM1of2 193	. . 3 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 ∧ ¬ 𝜓)) → (𝜑 → 𝜓))
4	2, 3	ndime-P3.6 171	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 ∧ ¬ 𝜓)) → 𝜓)
5	1	ndandel-P3.8 173	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 ∧ ¬ 𝜓)) → ¬ 𝜓)
6	4, 5	rcp-NDNEGI2 219	1 ⊢ ((𝜑 → 𝜓) → ¬ (𝜑 ∧ ¬ 𝜓))

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ∧ wff-and 132

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-true-D2.4 155

This theorem is referenced by: (None)