Theorem ioran 764

Description: Negated disjunction in terms of conjunction. This version of DeMorgan's law is a biconditional for all propositions (not just decidable ones), unlike oranim 793, anordc 969, or ianordc 911. Compare Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.)

Assertion

Ref	Expression
ioran	⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))

Proof of Theorem ioran

Step	Hyp	Ref	Expression
1		pm2.45 750	. . 3 ⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜑)
2		pm2.46 751	. . 3 ⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜓)
3	1, 2	jca 306	. 2 ⊢ (¬ (𝜑 ∨ 𝜓) → (¬ 𝜑 ∧ ¬ 𝜓))
4		simpl 109	. . . . 5 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ 𝜑)
5	4	con2i 636	. . . 4 ⊢ (𝜑 → ¬ (¬ 𝜑 ∧ ¬ 𝜓))
6		simpr 110	. . . . 5 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ 𝜓)
7	6	con2i 636	. . . 4 ⊢ (𝜓 → ¬ (¬ 𝜑 ∧ ¬ 𝜓))
8	5, 7	jaoi 728	. . 3 ⊢ ((𝜑 ∨ 𝜓) → ¬ (¬ 𝜑 ∧ ¬ 𝜓))
9	8	con2i 636	. 2 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 ∨ 𝜓))
10	3, 9	impbii 126	1 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 720

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721

This proof depends on definitions:  df-bi 117

This theorem is used by:  pm4.56  792  nnexmid  862  dcor  948  3ioran  1024  3ori  1341  ecase2d  1392  unssdif  3466  difundi  3483  dcun  3637  sotricim  4470  sotritrieq  4472  en2lp  4703  poxp  6472  nntri2  6772  finexdc  7213  elssdc  7215  unfidisj  7235  fidcenumlemrks  7277  pw1nel3  7596  sucpw1nel3  7598  onntri45  7606  aptipr  8014  lttri3  8411  letr  8414  apirr  8941  apti  8958  elnnz  9663  xrlttri3  10215  xrletr  10226  exp3val  10998  bcval4  11211  hashunlem  11265  maxleast  12001  xrmaxlesup  12049  lcmval  12865  lcmcllem  12869  lcmgcdlem  12879  isprm3  12920  pcpremul  13100  ivthinc  15886  lgsdir2  16374  2lgslem3  16442  structiedg0val  16503  vtxd0nedgbfi  16762  vdegp1aid  16777  bj-nnor  16984  pwtrufal  17249  pwle2  17250