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Mirrors > Home > ILE Home > Th. List > ioran | GIF version |
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 755, anordc 925, or ianordc 869. Compare Theorem *4.56 of [WhiteheadRussell] p. 120. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 31-Jan-2015.) |
Ref | Expression |
---|---|
ioran | ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.45 712 | . . 3 ⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜑) | |
2 | pm2.46 713 | . . 3 ⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜓) | |
3 | 1, 2 | jca 304 | . 2 ⊢ (¬ (𝜑 ∨ 𝜓) → (¬ 𝜑 ∧ ¬ 𝜓)) |
4 | simpl 108 | . . . . 5 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ 𝜑) | |
5 | 4 | con2i 601 | . . . 4 ⊢ (𝜑 → ¬ (¬ 𝜑 ∧ ¬ 𝜓)) |
6 | simpr 109 | . . . . 5 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ 𝜓) | |
7 | 6 | con2i 601 | . . . 4 ⊢ (𝜓 → ¬ (¬ 𝜑 ∧ ¬ 𝜓)) |
8 | 5, 7 | jaoi 690 | . . 3 ⊢ ((𝜑 ∨ 𝜓) → ¬ (¬ 𝜑 ∧ ¬ 𝜓)) |
9 | 8 | con2i 601 | . 2 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 ∨ 𝜓)) |
10 | 3, 9 | impbii 125 | 1 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 ∨ wo 682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: pm4.56 754 nnexmid 820 dcor 904 3ioran 962 3ori 1263 unssdif 3281 difundi 3298 dcun 3443 sotricim 4215 sotritrieq 4217 en2lp 4439 poxp 6097 nntri2 6358 finexdc 6764 unfidisj 6778 fidcenumlemrks 6809 aptipr 7417 lttri3 7812 letr 7815 apirr 8335 apti 8352 elnnz 9032 xrlttri3 9551 xrletr 9559 exp3val 10263 bcval4 10466 hashunlem 10518 maxleast 10953 xrmaxlesup 10996 lcmval 11671 lcmcllem 11675 lcmgcdlem 11685 isprm3 11726 ivthinc 12717 bj-nnor 12873 pwtrufal 13119 pwle2 13120 |
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