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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 598 | . . . 4 ⊢ (𝜑 → (𝜓 → ¬ (𝜑 → ¬ 𝜓))) | |
2 | 1 | imp 122 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → ¬ (𝜑 → ¬ 𝜓)) |
3 | 2 | con2i 589 | . 2 ⊢ ((𝜑 → ¬ 𝜓) → ¬ (𝜑 ∧ 𝜓)) |
4 | pm3.2 137 | . . 3 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓))) | |
5 | 4 | con3rr3 595 | . 2 ⊢ (¬ (𝜑 ∧ 𝜓) → (𝜑 → ¬ 𝜓)) |
6 | 3, 5 | impbii 124 | 1 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 |
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 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: imnani 657 nan 658 pm3.24 659 imandc 819 ianordc 832 pm5.17dc 843 dn1dc 901 xorbin 1315 xordc1 1324 alinexa 1534 ralinexa 2393 rabeq0 3274 disj 3292 minel 3305 disjsn 3454 sotricim 4078 poirr2 4737 funun 4964 imadiflem 4998 imadif 4999 brprcneu 5191 prltlu 6677 caucvgprlemnbj 6857 caucvgprprlemnbj 6883 xrltnsym2 8869 fzp1nel 9121 |
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