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Mirrors > Home > ILE Home > Th. List > 3pm3.2i | GIF version |
Description: Infer conjunction of premises. (Contributed by NM, 10-Feb-1995.) |
Ref | Expression |
---|---|
3pm3.2i.1 | ⊢ 𝜑 |
3pm3.2i.2 | ⊢ 𝜓 |
3pm3.2i.3 | ⊢ 𝜒 |
Ref | Expression |
---|---|
3pm3.2i | ⊢ (𝜑 ∧ 𝜓 ∧ 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3pm3.2i.1 | . . 3 ⊢ 𝜑 | |
2 | 3pm3.2i.2 | . . 3 ⊢ 𝜓 | |
3 | 1, 2 | pm3.2i 272 | . 2 ⊢ (𝜑 ∧ 𝜓) |
4 | 3pm3.2i.3 | . 2 ⊢ 𝜒 | |
5 | df-3an 980 | . 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) | |
6 | 3, 4, 5 | mpbir2an 942 | 1 ⊢ (𝜑 ∧ 𝜓 ∧ 𝜒) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∧ w3a 978 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 df-3an 980 |
This theorem is referenced by: mpbir3an 1179 3jaoi 1303 ftp 5701 4bc2eq6 10749 halfleoddlt 11893 strleun 12557 strle1g 12559 slotstnscsi 12644 slotsdnscsi 12668 slotsdifunifndx 12677 2irrexpqap 14327 lgslem2 14333 lgsdir2lem2 14361 lgsdir2lem3 14362 ex-dvds 14402 nconstwlpolem0 14730 |
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