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Mirrors > Home > ILE Home > Th. List > bianfd | GIF version |
Description: A wff conjoined with falsehood is false. (Contributed by NM, 27-Mar-1995.) (Proof shortened by Wolf Lammen, 5-Nov-2013.) |
Ref | Expression |
---|---|
bianfd.1 | ⊢ (𝜑 → ¬ 𝜓) |
Ref | Expression |
---|---|
bianfd | ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bianfd.1 | . 2 ⊢ (𝜑 → ¬ 𝜓) | |
2 | 1 | intnanrd 918 | . 2 ⊢ (𝜑 → ¬ (𝜓 ∧ 𝜒)) |
3 | 1, 2 | 2falsed 692 | 1 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜒))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: eueq2dc 2881 eueq3dc 2882 |
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