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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 840, anordc 897, or ianordc 832. 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 | pm2.45 689 | . . 3 ⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜑) | |
2 | pm2.46 690 | . . 3 ⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜓) | |
3 | 1, 2 | jca 300 | . 2 ⊢ (¬ (𝜑 ∨ 𝜓) → (¬ 𝜑 ∧ ¬ 𝜓)) |
4 | simpl 107 | . . . . 5 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ 𝜑) | |
5 | 4 | con2i 589 | . . . 4 ⊢ (𝜑 → ¬ (¬ 𝜑 ∧ ¬ 𝜓)) |
6 | simpr 108 | . . . . 5 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ 𝜓) | |
7 | 6 | con2i 589 | . . . 4 ⊢ (𝜓 → ¬ (¬ 𝜑 ∧ ¬ 𝜓)) |
8 | 5, 7 | jaoi 668 | . . 3 ⊢ ((𝜑 ∨ 𝜓) → ¬ (¬ 𝜑 ∧ ¬ 𝜓)) |
9 | 8 | con2i 589 | . 2 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → ¬ (𝜑 ∨ 𝜓)) |
10 | 3, 9 | impbii 124 | 1 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 102 ↔ wb 103 ∨ wo 661 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: pm4.56 839 dcor 876 3ioran 934 3ori 1231 unssdif 3199 difundi 3216 sotricim 4078 sotritrieq 4080 en2lp 4297 poxp 5873 nntri2 6096 aptipr 6831 lttri3 7191 letr 7194 apirr 7705 apti 7722 elnnz 8361 xrlttri3 8872 xrletr 8878 expival 9478 bcval4 9679 maxleast 10099 lcmval 10445 lcmcllem 10449 lcmgcdlem 10459 isprm3 10500 nnexmid 10570 |
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