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Mirrors > Home > MPE Home > Th. List > syl331anc | Structured version Visualization version GIF version |
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) |
Ref | Expression |
---|---|
syl3anc.1 | ⊢ (𝜑 → 𝜓) |
syl3anc.2 | ⊢ (𝜑 → 𝜒) |
syl3anc.3 | ⊢ (𝜑 → 𝜃) |
syl3Xanc.4 | ⊢ (𝜑 → 𝜏) |
syl23anc.5 | ⊢ (𝜑 → 𝜂) |
syl33anc.6 | ⊢ (𝜑 → 𝜁) |
syl133anc.7 | ⊢ (𝜑 → 𝜎) |
syl331anc.8 | ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁) ∧ 𝜎) → 𝜌) |
Ref | Expression |
---|---|
syl331anc | ⊢ (𝜑 → 𝜌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl3anc.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | syl3anc.2 | . 2 ⊢ (𝜑 → 𝜒) | |
3 | syl3anc.3 | . 2 ⊢ (𝜑 → 𝜃) | |
4 | syl3Xanc.4 | . . 3 ⊢ (𝜑 → 𝜏) | |
5 | syl23anc.5 | . . 3 ⊢ (𝜑 → 𝜂) | |
6 | syl33anc.6 | . . 3 ⊢ (𝜑 → 𝜁) | |
7 | 4, 5, 6 | 3jca 1159 | . 2 ⊢ (𝜑 → (𝜏 ∧ 𝜂 ∧ 𝜁)) |
8 | syl133anc.7 | . 2 ⊢ (𝜑 → 𝜎) | |
9 | syl331anc.8 | . 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁) ∧ 𝜎) → 𝜌) | |
10 | 1, 2, 3, 7, 8, 9 | syl311anc 1504 | 1 ⊢ (𝜑 → 𝜌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1108 |
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 199 df-an 386 df-3an 1110 |
This theorem is referenced by: syl332anc 1521 syl333anc 1522 qredeu 15703 brbtwn2 26134 3atlem4 35499 3atlem6 35501 llnexchb2 35882 osumcllem9N 35977 cdlemd4 36214 cdleme26fALTN 36375 cdleme26f 36376 cdleme36m 36474 cdlemg17b 36675 cdlemg17h 36681 cdlemk38 36928 cdlemk53b 36969 cdlemkyyN 36975 cdlemk43N 36976 |
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