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Mirrors > Home > ILE Home > Th. List > a1bi | GIF version |
Description: Inference 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 240 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝜓 ↔ (𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ 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 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: mt2bi 679 truimfal 1405 equsal 1720 equveli 1752 equsalv 1786 ralv 2747 relop 4759 |
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