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Mirrors > Home > ILE Home > Th. List > a1bi | GIF version |
Description: Inference rule introducing a theorem as an antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Nov-2012.) |
Ref | Expression |
---|---|
a1bi.1 | ⊢ 𝜑 |
Ref | Expression |
---|---|
a1bi | ⊢ (𝜓 ↔ (𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | a1bi.1 | . 2 ⊢ 𝜑 | |
2 | biimt 239 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓))) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (𝜓 ↔ (𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ 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 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: mt2bi 641 truimfal 1341 equsal 1655 equveli 1682 ralv 2616 relop 4504 |
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