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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 | pm3.21 261 | . 2 ⊢ (𝜑 → (𝜓 → (𝜓 ∧ 𝜑))) | |
2 | simpl 108 | . 2 ⊢ ((𝜓 ∧ 𝜑) → 𝜓) | |
3 | 1, 2 | impbid1 141 | 1 ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜑))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: biantru 297 biantrud 299 ancrb 316 rbaibd 874 dedlem0a 917 fvopab6 5435 fressnfv 5523 tpostpos 6067 nnmword 6317 unfiexmid 6708 ltmpig 6995 sup3exmid 8515 xrmaxiflemcom 10808 |
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