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Mirrors > Home > MPE Home > Th. List > syl212anc | Structured version Visualization version GIF version |
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) |
Ref | Expression |
---|---|
syl12anc.1 | ⊢ (𝜑 → 𝜓) |
syl12anc.2 | ⊢ (𝜑 → 𝜒) |
syl12anc.3 | ⊢ (𝜑 → 𝜃) |
syl22anc.4 | ⊢ (𝜑 → 𝜏) |
syl23anc.5 | ⊢ (𝜑 → 𝜂) |
syl212anc.6 | ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃 ∧ (𝜏 ∧ 𝜂)) → 𝜁) |
Ref | Expression |
---|---|
syl212anc | ⊢ (𝜑 → 𝜁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl12anc.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | syl12anc.2 | . 2 ⊢ (𝜑 → 𝜒) | |
3 | syl12anc.3 | . 2 ⊢ (𝜑 → 𝜃) | |
4 | syl22anc.4 | . . 3 ⊢ (𝜑 → 𝜏) | |
5 | syl23anc.5 | . . 3 ⊢ (𝜑 → 𝜂) | |
6 | 4, 5 | jca 553 | . 2 ⊢ (𝜑 → (𝜏 ∧ 𝜂)) |
7 | syl212anc.6 | . 2 ⊢ (((𝜓 ∧ 𝜒) ∧ 𝜃 ∧ (𝜏 ∧ 𝜂)) → 𝜁) | |
8 | 1, 2, 3, 6, 7 | syl211anc 1324 | 1 ⊢ (𝜑 → 𝜁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 |
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 196 df-an 385 df-3an 1033 |
This theorem is referenced by: pntrmax 25053 tglineineq 25338 tglineinteq 25340 paddasslem4 34127 4atexlemu 34368 4atexlemv 34369 cdleme20aN 34615 cdleme20g 34621 cdlemg9a 34938 cdlemg12a 34949 cdlemg17dALTN 34970 cdlemg18b 34985 cdlemg18c 34986 |
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