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Mirrors > Home > ILE Home > Th. List > 2false | GIF version |
Description: Two falsehoods are equivalent. (Contributed by NM, 4-Apr-2005.) (Revised by Mario Carneiro, 31-Jan-2015.) |
Ref | Expression |
---|---|
2false.1 | ⊢ ¬ 𝜑 |
2false.2 | ⊢ ¬ 𝜓 |
Ref | Expression |
---|---|
2false | ⊢ (𝜑 ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2false.1 | . . 3 ⊢ ¬ 𝜑 | |
2 | 1 | pm2.21i 608 | . 2 ⊢ (𝜑 → 𝜓) |
3 | 2false.2 | . . 3 ⊢ ¬ 𝜓 | |
4 | 3 | pm2.21i 608 | . 2 ⊢ (𝜓 → 𝜑) |
5 | 2, 4 | impbii 124 | 1 ⊢ (𝜑 ↔ 𝜓) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 103 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia2 105 ax-ia3 106 ax-in2 578 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: bianfi 891 bifal 1300 dfnul2 3274 dfnul3 3275 rab0 3297 iun0 3763 0iun 3764 0xp 4479 cnv0 4792 co02 4901 0er 6259 bdnth 11082 bdnthALT 11083 |
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