Theorem xordidc 1290

Description: Conjunction distributes over exclusive-or, for decidable propositions. This is one way to interpret the distributive law of multiplication over addition in modulo 2 arithmetic. (Contributed by Jim Kingdon, 14-Jul-2018.)

Assertion

Ref	Expression
xordidc	⊢ (DECID 𝜑 → (DECID 𝜓 → (DECID 𝜒 → ((𝜑 ∧ (𝜓 ⊻ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ⊻ (𝜑 ∧ 𝜒))))))

Proof of Theorem xordidc

Step	Hyp	Ref	Expression
1		dcbi 844	. . . . 5 ⊢ (DECID 𝜓 → (DECID 𝜒 → DECID (𝜓 ↔ 𝜒)))
2	1	imp 115	. . . 4 ⊢ ((DECID 𝜓 ∧ DECID 𝜒) → DECID (𝜓 ↔ 𝜒))
3		annimdc 845	. . . . . 6 ⊢ (DECID 𝜑 → (DECID (𝜓 ↔ 𝜒) → ((𝜑 ∧ ¬ (𝜓 ↔ 𝜒)) ↔ ¬ (𝜑 → (𝜓 ↔ 𝜒)))))
4	3	imp 115	. . . . 5 ⊢ ((DECID 𝜑 ∧ DECID (𝜓 ↔ 𝜒)) → ((𝜑 ∧ ¬ (𝜓 ↔ 𝜒)) ↔ ¬ (𝜑 → (𝜓 ↔ 𝜒))))
5		pm5.32 426	. . . . . 6 ⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)))
6	5	notbii 594	. . . . 5 ⊢ (¬ (𝜑 → (𝜓 ↔ 𝜒)) ↔ ¬ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)))
7	4, 6	syl6bb 185	. . . 4 ⊢ ((DECID 𝜑 ∧ DECID (𝜓 ↔ 𝜒)) → ((𝜑 ∧ ¬ (𝜓 ↔ 𝜒)) ↔ ¬ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))))
8	2, 7	sylan2 270	. . 3 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ DECID 𝜒)) → ((𝜑 ∧ ¬ (𝜓 ↔ 𝜒)) ↔ ¬ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))))
9		xornbidc 1282	. . . . . 6 ⊢ (DECID 𝜓 → (DECID 𝜒 → ((𝜓 ⊻ 𝜒) ↔ ¬ (𝜓 ↔ 𝜒))))
10	9	imp 115	. . . . 5 ⊢ ((DECID 𝜓 ∧ DECID 𝜒) → ((𝜓 ⊻ 𝜒) ↔ ¬ (𝜓 ↔ 𝜒)))
11	10	adantl 262	. . . 4 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ DECID 𝜒)) → ((𝜓 ⊻ 𝜒) ↔ ¬ (𝜓 ↔ 𝜒)))
12	11	anbi2d 437	. . 3 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ DECID 𝜒)) → ((𝜑 ∧ (𝜓 ⊻ 𝜒)) ↔ (𝜑 ∧ ¬ (𝜓 ↔ 𝜒))))
13		dcan 842	. . . . . 6 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (𝜑 ∧ 𝜓)))
14	13	imp 115	. . . . 5 ⊢ ((DECID 𝜑 ∧ DECID 𝜓) → DECID (𝜑 ∧ 𝜓))
15	14	adantrr 448	. . . 4 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ DECID 𝜒)) → DECID (𝜑 ∧ 𝜓))
16		dcan 842	. . . . . 6 ⊢ (DECID 𝜑 → (DECID 𝜒 → DECID (𝜑 ∧ 𝜒)))
17	16	imp 115	. . . . 5 ⊢ ((DECID 𝜑 ∧ DECID 𝜒) → DECID (𝜑 ∧ 𝜒))
18	17	adantrl 447	. . . 4 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ DECID 𝜒)) → DECID (𝜑 ∧ 𝜒))
19		xornbidc 1282	. . . 4 ⊢ (DECID (𝜑 ∧ 𝜓) → (DECID (𝜑 ∧ 𝜒) → (((𝜑 ∧ 𝜓) ⊻ (𝜑 ∧ 𝜒)) ↔ ¬ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)))))
20	15, 18, 19	sylc 56	. . 3 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ DECID 𝜒)) → (((𝜑 ∧ 𝜓) ⊻ (𝜑 ∧ 𝜒)) ↔ ¬ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒))))
21	8, 12, 20	3bitr4d 209	. 2 ⊢ ((DECID 𝜑 ∧ (DECID 𝜓 ∧ DECID 𝜒)) → ((𝜑 ∧ (𝜓 ⊻ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ⊻ (𝜑 ∧ 𝜒))))
22	21	exp32 347	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → (DECID 𝜒 → ((𝜑 ∧ (𝜓 ⊻ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ⊻ (𝜑 ∧ 𝜒))))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  DECID wdc 742   ⊻ wxo 1266

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743  df-xor 1267

This theorem is referenced by: (None)