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  3439  difundi  3456  dcun  3601  sotricim  4413  sotritrieq  4415  en2lp  4645  poxp  6376  nntri2  6638  finexdc  7060  unfidisj  7080  fidcenumlemrks  7116  pw1nel3  7412  sucpw1nel3  7414  onntri45  7422  aptipr  7824  lttri3  8222  letr  8225  apirr  8748  apti  8765  elnnz  9452  xrlttri3  9989  xrletr  10000  exp3val  10758  bcval4  10969  hashunlem  11021  maxleast  11719  xrmaxlesup  11765  lcmval  12580  lcmcllem  12584  lcmgcdlem  12594  isprm3  12635  pcpremul  12811  ivthinc  15311  lgsdir2  15706  2lgslem3  15774  structiedg0val  15835  bj-nnor  16056  pwtrufal  16322  pwle2  16323