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Mirrors > Home > ILE Home > Th. List > imnan | GIF version |
Description: Express implication in terms of conjunction. (Contributed by NM, 9-Apr-1994.) (Revised by Mario Carneiro, 1-Feb-2015.) |
Ref | Expression |
---|---|
imnan | ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.2im 638 | . . . 4 ⊢ (𝜑 → (𝜓 → ¬ (𝜑 → ¬ 𝜓))) | |
2 | 1 | imp 124 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → ¬ (𝜑 → ¬ 𝜓)) |
3 | 2 | con2i 628 | . 2 ⊢ ((𝜑 → ¬ 𝜓) → ¬ (𝜑 ∧ 𝜓)) |
4 | pm3.2 139 | . . 3 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓))) | |
5 | 4 | con3rr3 634 | . 2 ⊢ (¬ (𝜑 ∧ 𝜓) → (𝜑 → ¬ 𝜓)) |
6 | 3, 5 | impbii 126 | 1 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 |
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 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: imnani 692 nan 693 pm3.24 694 imanst 889 ianordc 900 pm5.17dc 905 dn1dc 961 xorbin 1394 xordc1 1403 alinexa 1613 dfrex2dc 2478 ralinexa 2514 rabeq0 3464 disj 3483 minel 3496 disjsn 3666 sotricim 4335 poirr2 5033 funun 5272 imadiflem 5307 imadif 5308 brprcneu 5520 2omotaplemap 7270 prltlu 7500 caucvgprlemnbj 7680 caucvgprprlemnbj 7706 suplocexprlemmu 7731 xrltnsym2 9808 fzp1nel 10118 fsumsplit 11429 sumsplitdc 11454 phiprmpw 12236 odzdvds 12259 pcdvdsb 12333 lgsne0 14735 bj-nnor 14782 |
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