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Mirrors > Home > MPE Home > Th. List > abai | Structured version Visualization version GIF version |
Description: Introduce one conjunct as an antecedent to the other. "abai" stands for "and, biconditional, and, implication". (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Dec-2012.) |
Ref | Expression |
---|---|
abai | ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ (𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biimt 360 | . 2 ⊢ (𝜑 → (𝜓 ↔ (𝜑 → 𝜓))) | |
2 | 1 | pm5.32i 575 | 1 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ (𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 206 df-an 397 |
This theorem is referenced by: pm5.75 1027 exintrbi 1894 dfeumo 2531 eu6 2568 dfeu 2589 r19.29imd 3118 dfss4 4257 indifdi 4282 choc0 30566 |
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