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Mirrors > Home > ILE Home > Th. List > biimpac | GIF version |
Description: Inference from a logical equivalence. (Contributed by NM, 3-May-1994.) |
Ref | Expression |
---|---|
biimpa.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
biimpac | ⊢ ((𝜓 ∧ 𝜑) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biimpa.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | biimpcd 159 | . 2 ⊢ (𝜓 → (𝜑 → 𝜒)) |
3 | 2 | imp 124 | 1 ⊢ ((𝜓 ∧ 𝜑) → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: gencbvex2 2784 ordtri2or2exmidlem 4521 onsucelsucexmidlem 4524 ordsuc 4558 onsucuni2 4559 poltletr 5024 tz6.12-1 5537 nfunsn 5544 nnaordex 6522 th3qlem1 6630 ssfilem 6868 diffitest 6880 nqnq0pi 7415 distrlem1prl 7559 distrlem1pru 7560 eqle 8026 flodddiv4 11909 zabsle1 14033 |
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