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Mirrors > Home > ILE Home > Th. List > niabn | GIF version |
Description: Miscellaneous inference relating falsehoods. (Contributed by NM, 31-Mar-1994.) |
Ref | Expression |
---|---|
niabn.1 | ⊢ 𝜑 |
Ref | Expression |
---|---|
niabn | ⊢ (¬ 𝜓 → ((𝜒 ∧ 𝜓) ↔ ¬ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 103 | . 2 ⊢ ((𝜒 ∧ 𝜓) → 𝜓) | |
2 | niabn.1 | . . 3 ⊢ 𝜑 | |
3 | 2 | pm2.24i 553 | . 2 ⊢ (¬ 𝜑 → 𝜓) |
4 | 1, 3 | pm5.21ni 619 | 1 ⊢ (¬ 𝜓 → ((𝜒 ∧ 𝜓) ↔ ¬ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ↔ wb 98 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: ninba 879 |
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