Theorem pm5.18dc 776

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 742	. 2 ⊢ (DECID φ ↔ (φ ∨ ¬ φ))
2		pm5.501 233	. . . . . . . 8 ⊢ (φ → (¬ ψ ↔ (φ ↔ ¬ ψ)))
3	2	a1d 22	. . . . . . 7 ⊢ (φ → (DECID ψ → (¬ ψ ↔ (φ ↔ ¬ ψ))))
4	3	con1biddc 769	. . . . . 6 ⊢ (φ → (DECID ψ → (¬ (φ ↔ ¬ ψ) ↔ ψ)))
5	4	imp 115	. . . . 5 ⊢ ((φ ∧ DECID ψ) → (¬ (φ ↔ ¬ ψ) ↔ ψ))
6		pm5.501 233	. . . . . 6 ⊢ (φ → (ψ ↔ (φ ↔ ψ)))
7	6	adantr 261	. . . . 5 ⊢ ((φ ∧ DECID ψ) → (ψ ↔ (φ ↔ ψ)))
8	5, 7	bitr2d 178	. . . 4 ⊢ ((φ ∧ DECID ψ) → ((φ ↔ ψ) ↔ ¬ (φ ↔ ¬ ψ)))
9	8	ex 108	. . 3 ⊢ (φ → (DECID ψ → ((φ ↔ ψ) ↔ ¬ (φ ↔ ¬ ψ))))
10		dcn 745	. . . . . . 7 ⊢ (DECID ψ → DECID ¬ ψ)
11		nbn2 612	. . . . . . . . 9 ⊢ (¬ φ → (¬ ¬ ψ ↔ (φ ↔ ¬ ψ)))
12	11	a1d 22	. . . . . . . 8 ⊢ (¬ φ → (DECID ¬ ψ → (¬ ¬ ψ ↔ (φ ↔ ¬ ψ))))
13	12	con1biddc 769	. . . . . . 7 ⊢ (¬ φ → (DECID ¬ ψ → (¬ (φ ↔ ¬ ψ) ↔ ¬ ψ)))
14	10, 13	syl5 28	. . . . . 6 ⊢ (¬ φ → (DECID ψ → (¬ (φ ↔ ¬ ψ) ↔ ¬ ψ)))
15	14	imp 115	. . . . 5 ⊢ ((¬ φ ∧ DECID ψ) → (¬ (φ ↔ ¬ ψ) ↔ ¬ ψ))
16		nbn2 612	. . . . . 6 ⊢ (¬ φ → (¬ ψ ↔ (φ ↔ ψ)))
17	16	adantr 261	. . . . 5 ⊢ ((¬ φ ∧ DECID ψ) → (¬ ψ ↔ (φ ↔ ψ)))
18	15, 17	bitr2d 178	. . . 4 ⊢ ((¬ φ ∧ DECID ψ) → ((φ ↔ ψ) ↔ ¬ (φ ↔ ¬ ψ)))
19	18	ex 108	. . 3 ⊢ (¬ φ → (DECID ψ → ((φ ↔ ψ) ↔ ¬ (φ ↔ ¬ ψ))))
20	9, 19	jaoi 635	. 2 ⊢ ((φ ∨ ¬ φ) → (DECID ψ → ((φ ↔ ψ) ↔ ¬ (φ ↔ ¬ ψ))))
21	1, 20	sylbi 114	1 ⊢ (DECID φ → (DECID ψ → ((φ ↔ ψ) ↔ ¬ (φ ↔ ¬ ψ))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628  DECID wdc 741

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

This theorem is referenced by:  xor3dc  1275  dfbi3dc  1285