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Mirrors > Home > ILE Home > Th. List > iba | GIF version |
Description: Introduction of antecedent as conjunct. Theorem *4.73 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Mar-1994.) (Revised by NM, 24-Mar-2013.) |
Ref | Expression |
---|---|
iba | ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.21 260 | . 2 ⊢ (𝜑 → (𝜓 → (𝜓 ∧ 𝜑))) | |
2 | simpl 107 | . 2 ⊢ ((𝜓 ∧ 𝜑) → 𝜓) | |
3 | 1, 2 | impbid1 140 | 1 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜑))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ 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: biantru 296 biantrud 298 ancrb 315 rbaibd 866 dedlem0a 909 fvopab6 5285 fressnfv 5371 tpostpos 5902 nnmword 6114 ltmpig 6529 |
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