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Mirrors > Home > ILE Home > Th. List > syl122anc | GIF version |
Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.) |
Ref | Expression |
---|---|
sylXanc.1 | ⊢ (𝜑 → 𝜓) |
sylXanc.2 | ⊢ (𝜑 → 𝜒) |
sylXanc.3 | ⊢ (𝜑 → 𝜃) |
sylXanc.4 | ⊢ (𝜑 → 𝜏) |
sylXanc.5 | ⊢ (𝜑 → 𝜂) |
syl122anc.6 | ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂)) → 𝜁) |
Ref | Expression |
---|---|
syl122anc | ⊢ (𝜑 → 𝜁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylXanc.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | sylXanc.2 | . 2 ⊢ (𝜑 → 𝜒) | |
3 | sylXanc.3 | . 2 ⊢ (𝜑 → 𝜃) | |
4 | sylXanc.4 | . . 3 ⊢ (𝜑 → 𝜏) | |
5 | sylXanc.5 | . . 3 ⊢ (𝜑 → 𝜂) | |
6 | 4, 5 | jca 300 | . 2 ⊢ (𝜑 → (𝜏 ∧ 𝜂)) |
7 | syl122anc.6 | . 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂)) → 𝜁) | |
8 | 1, 2, 3, 6, 7 | syl121anc 1174 | 1 ⊢ (𝜑 → 𝜁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∧ w3a 919 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 |
This theorem depends on definitions: df-bi 115 df-3an 921 |
This theorem is referenced by: divdiv32apd 7902 divcanap5d 7903 divcanap7d 7905 divdivap1d 7908 divdivap2d 7909 cau3lem 10000 prmind2 10502 |
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