Theorem dfandc 879

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 632. (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 632	. . . 4 ⊢ (𝜑 → (𝜓 → ¬ (𝜑 → ¬ 𝜓)))
2	1	imp 123	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ¬ (𝜑 → ¬ 𝜓))
3		simplimdc 855	. . . . . . 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 854	. . . . . . 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 829

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 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830

This theorem is referenced by:  pm4.63dc  881  pm4.54dc  897