Theorem xordidc 1287

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 843	. . . . 5 ⊢ (DECID ψ → (DECID χ → DECID (ψ ↔ χ)))
2	1	imp 115	. . . 4 ⊢ ((DECID ψ ∧ DECID χ) → DECID (ψ ↔ χ))
3		annimdc 844	. . . . . 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 593	. . . . 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 1279	. . . . . 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 841	. . . . . 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 841	. . . . . 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 1279	. . . 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 741   ⊻ wxo 1265

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 629

This theorem depends on definitions:  df-bi 110  df-dc 742  df-xor 1266

This theorem is referenced by: (None)