Theorem pm5.17dc 810

Description: Two ways of stating exclusive-or which are equivalent for a decidable proposition. Based on theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 16-Apr-2018.)

Assertion

Ref	Expression
pm5.17dc	⊢ (DECID 𝜓 → (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ↔ ¬ 𝜓)))

Proof of Theorem pm5.17dc

Step	Hyp	Ref	Expression
1		bicom 128	. 2 ⊢ ((𝜑 ↔ ¬ 𝜓) ↔ (¬ 𝜓 ↔ 𝜑))
2		dfbi2 368	. . 3 ⊢ ((¬ 𝜓 ↔ 𝜑) ↔ ((¬ 𝜓 → 𝜑) ∧ (𝜑 → ¬ 𝜓)))
3		orcom 647	. . . . 5 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑))
4		dfordc 791	. . . . 5 ⊢ (DECID 𝜓 → ((𝜓 ∨ 𝜑) ↔ (¬ 𝜓 → 𝜑)))
5	3, 4	syl5rbb 182	. . . 4 ⊢ (DECID 𝜓 → ((¬ 𝜓 → 𝜑) ↔ (𝜑 ∨ 𝜓)))
6		imnan 624	. . . . 5 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
7	6	a1i 9	. . . 4 ⊢ (DECID 𝜓 → ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)))
8	5, 7	anbi12d 442	. . 3 ⊢ (DECID 𝜓 → (((¬ 𝜓 → 𝜑) ∧ (𝜑 → ¬ 𝜓)) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))))
9	2, 8	syl5bb 181	. 2 ⊢ (DECID 𝜓 → ((¬ 𝜓 ↔ 𝜑) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))))
10	1, 9	syl5rbb 182	1 ⊢ (DECID 𝜓 → (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ↔ ¬ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  DECID wdc 742

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

This theorem is referenced by:  xor2dc  1281