Theorem ioran 757

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 786, anordc 962, or ianordc 904. 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		pm2.45 743	. . 3 ⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜑)
2		pm2.46 744	. . 3 ⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜓)
3	1, 2	jca 306	. 2 ⊢ (¬ (𝜑 ∨ 𝜓) → (¬ 𝜑 ∧ ¬ 𝜓))
4		simpl 109	. . . . 5 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ 𝜑)
5	4	con2i 630	. . . 4 ⊢ (𝜑 → ¬ (¬ 𝜑 ∧ ¬ 𝜓))
6		simpr 110	. . . . 5 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ 𝜓)
7	6	con2i 630	. . . 4 ⊢ (𝜓 → ¬ (¬ 𝜑 ∧ ¬ 𝜓))
8	5, 7	jaoi 721	. . 3 ⊢ ((𝜑 ∨ 𝜓) → ¬ (¬ 𝜑 ∧ ¬ 𝜓))
9	8	con2i 630	. 2 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 ∨ 𝜓))
10	3, 9	impbii 126	1 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))

Syntax hints:  ¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 713

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 617  ax-in2 618  ax-io 714

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm4.56  785  nnexmid  855  dcor  941  3ioran  1017  3ori  1334  ecase2d  1385  unssdif  3440  difundi  3457  dcun  3602  sotricim  4418  sotritrieq  4420  en2lp  4650  poxp  6392  nntri2  6657  finexdc  7085  elssdc  7087  unfidisj  7107  fidcenumlemrks  7143  pw1nel3  7439  sucpw1nel3  7441  onntri45  7449  aptipr  7851  lttri3  8249  letr  8252  apirr  8775  apti  8792  elnnz  9479  xrlttri3  10022  xrletr  10033  exp3val  10793  bcval4  11004  hashunlem  11057  maxleast  11764  xrmaxlesup  11810  lcmval  12625  lcmcllem  12629  lcmgcdlem  12639  isprm3  12680  pcpremul  12856  ivthinc  15357  lgsdir2  15752  2lgslem3  15820  structiedg0val  15881  vtxd0nedgbfi  16105  bj-nnor  16266  pwtrufal  16534  pwle2  16535