Theorem pm5.18dc 878

Description: Relationship between an equivalence and an equivalence with some negation, for decidable propositions. Based on theorem *5.18 of [WhiteheadRussell] p. 124. Given decidability, we can consider ¬ (𝜑 ↔ ¬ 𝜓) to represent "negated exclusive-or". (Contributed by Jim Kingdon, 4-Apr-2018.)

Assertion

Ref	Expression
pm5.18dc	⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))

Proof of Theorem pm5.18dc

Step	Hyp	Ref	Expression
1		df-dc 830	. 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑))
2		pm5.501 243	. . . . . . . 8 ⊢ (𝜑 → (¬ 𝜓 ↔ (𝜑 ↔ ¬ 𝜓)))
3	2	a1d 22	. . . . . . 7 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 ↔ (𝜑 ↔ ¬ 𝜓))))
4	3	con1biddc 871	. . . . . 6 ⊢ (𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ ¬ 𝜓) ↔ 𝜓)))
5	4	imp 123	. . . . 5 ⊢ ((𝜑 ∧ DECID 𝜓) → (¬ (𝜑 ↔ ¬ 𝜓) ↔ 𝜓))
6		pm5.501 243	. . . . . 6 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ↔ 𝜓)))
7	6	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ DECID 𝜓) → (𝜓 ↔ (𝜑 ↔ 𝜓)))
8	5, 7	bitr2d 188	. . . 4 ⊢ ((𝜑 ∧ DECID 𝜓) → ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓)))
9	8	ex 114	. . 3 ⊢ (𝜑 → (DECID 𝜓 → ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))
10		dcn 837	. . . . . . 7 ⊢ (DECID 𝜓 → DECID ¬ 𝜓)
11		nbn2 692	. . . . . . . . 9 ⊢ (¬ 𝜑 → (¬ ¬ 𝜓 ↔ (𝜑 ↔ ¬ 𝜓)))
12	11	a1d 22	. . . . . . . 8 ⊢ (¬ 𝜑 → (DECID ¬ 𝜓 → (¬ ¬ 𝜓 ↔ (𝜑 ↔ ¬ 𝜓))))
13	12	con1biddc 871	. . . . . . 7 ⊢ (¬ 𝜑 → (DECID ¬ 𝜓 → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ¬ 𝜓)))
14	10, 13	syl5 32	. . . . . 6 ⊢ (¬ 𝜑 → (DECID 𝜓 → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ¬ 𝜓)))
15	14	imp 123	. . . . 5 ⊢ ((¬ 𝜑 ∧ DECID 𝜓) → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ¬ 𝜓))
16		nbn2 692	. . . . . 6 ⊢ (¬ 𝜑 → (¬ 𝜓 ↔ (𝜑 ↔ 𝜓)))
17	16	adantr 274	. . . . 5 ⊢ ((¬ 𝜑 ∧ DECID 𝜓) → (¬ 𝜓 ↔ (𝜑 ↔ 𝜓)))
18	15, 17	bitr2d 188	. . . 4 ⊢ ((¬ 𝜑 ∧ DECID 𝜓) → ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓)))
19	18	ex 114	. . 3 ⊢ (¬ 𝜑 → (DECID 𝜓 → ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))
20	9, 19	jaoi 711	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → (DECID 𝜓 → ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))
21	1, 20	sylbi 120	1 ⊢ (DECID 𝜑 → (DECID 𝜓 → ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703  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:  xor3dc  1382  dfbi3dc  1392