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Mirrors > Home > ILE Home > Th. List > biadanii | GIF version |
Description: Inference associated with biadani 612. Add a conjunction to an equivalence. (Contributed by Jeff Madsen, 20-Jun-2011.) (Proof shortened by BJ, 4-Mar-2023.) |
Ref | Expression |
---|---|
biadani.1 | ⊢ (𝜑 → 𝜓) |
biadanii.2 | ⊢ (𝜓 → (𝜑 ↔ 𝜒)) |
Ref | Expression |
---|---|
biadanii | ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biadanii.2 | . 2 ⊢ (𝜓 → (𝜑 ↔ 𝜒)) | |
2 | biadani.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
3 | 2 | biadani 612 | . 2 ⊢ ((𝜓 → (𝜑 ↔ 𝜒)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜒))) |
4 | 1, 3 | mpbi 145 | 1 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → 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 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: ismhm 12730 iscn2 13333 |
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