Theorem ioran 753

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 782, anordc 958, or ianordc 900. 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		pm2.45 739	. . 3 ⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜑)
2		pm2.46 740	. . 3 ⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜓)
3	1, 2	jca 306	. 2 ⊢ (¬ (𝜑 ∨ 𝜓) → (¬ 𝜑 ∧ ¬ 𝜓))
4		simpl 109	. . . . 5 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ 𝜑)
5	4	con2i 628	. . . 4 ⊢ (𝜑 → ¬ (¬ 𝜑 ∧ ¬ 𝜓))
6		simpr 110	. . . . 5 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ 𝜓)
7	6	con2i 628	. . . 4 ⊢ (𝜓 → ¬ (¬ 𝜑 ∧ ¬ 𝜓))
8	5, 7	jaoi 717	. . 3 ⊢ ((𝜑 ∨ 𝜓) → ¬ (¬ 𝜑 ∧ ¬ 𝜓))
9	8	con2i 628	. 2 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 ∨ 𝜓))
10	3, 9	impbii 126	1 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm4.56  781  nnexmid  851  dcor  937  3ioran  995  3ori  1311  unssdif  3398  difundi  3415  dcun  3560  sotricim  4358  sotritrieq  4360  en2lp  4590  poxp  6290  nntri2  6552  finexdc  6963  unfidisj  6983  fidcenumlemrks  7019  pw1nel3  7298  sucpw1nel3  7300  onntri45  7308  aptipr  7708  lttri3  8106  letr  8109  apirr  8632  apti  8649  elnnz  9336  xrlttri3  9872  xrletr  9883  exp3val  10633  bcval4  10844  hashunlem  10896  maxleast  11378  xrmaxlesup  11424  lcmval  12231  lcmcllem  12235  lcmgcdlem  12245  isprm3  12286  pcpremul  12462  ivthinc  14879  lgsdir2  15274  2lgslem3  15342  bj-nnor  15380  pwtrufal  15642  pwle2  15643