Theorem dfandc 874

Description: Definition of 'and' in terms of negation and implication, for decidable propositions. The forward direction holds for all propositions, and can (basically) be found at pm3.2im 627. (Contributed by Jim Kingdon, 30-Apr-2018.)

Assertion

Ref	Expression
dfandc	⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))))

Proof of Theorem dfandc

Step	Hyp	Ref	Expression
1		pm3.2im 627	. . . 4 ⊢ (𝜑 → (𝜓 → ¬ (𝜑 → ¬ 𝜓)))
2	1	imp 123	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ¬ (𝜑 → ¬ 𝜓))
3		simplimdc 850	. . . . . . 7 ⊢ (DECID 𝜑 → (¬ (𝜑 → ¬ 𝜓) → 𝜑))
4	3	adantr 274	. . . . . 6 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 → ¬ 𝜓) → 𝜑))
5	4	imp 123	. . . . 5 ⊢ (((DECID 𝜑 ∧ DECID 𝜓) ∧ ¬ (𝜑 → ¬ 𝜓)) → 𝜑)
6		simprimdc 849	. . . . . . 7 ⊢ (DECID 𝜓 → (¬ (𝜑 → ¬ 𝜓) → 𝜓))
7	6	adantl 275	. . . . . 6 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 → ¬ 𝜓) → 𝜓))
8	7	imp 123	. . . . 5 ⊢ (((DECID 𝜑 ∧ DECID 𝜓) ∧ ¬ (𝜑 → ¬ 𝜓)) → 𝜓)
9	5, 8	jca 304	. . . 4 ⊢ (((DECID 𝜑 ∧ DECID 𝜓) ∧ ¬ (𝜑 → ¬ 𝜓)) → (𝜑 ∧ 𝜓))
10	9	ex 114	. . 3 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 → ¬ 𝜓) → (𝜑 ∧ 𝜓)))
11	2, 10	impbid2 142	. 2 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓)))
12	11	ex 114	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825

This theorem is referenced by:  pm4.63dc  876  pm4.54dc  892