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Mirrors > Home > ILE Home > Th. List > syl32anc | 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) |
syl32anc.6 | ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂)) → 𝜁) |
Ref | Expression |
---|---|
syl32anc | ⊢ (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 304 | . 2 ⊢ (𝜑 → (𝜏 ∧ 𝜂)) |
7 | syl32anc.6 | . 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂)) → 𝜁) | |
8 | 1, 2, 3, 6, 7 | syl31anc 1219 | 1 ⊢ (𝜑 → 𝜁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 df-3an 964 |
This theorem is referenced by: ioom 10031 modifeq2int 10152 modaddmodup 10153 seq3f1olemqsum 10266 seq3f1o 10270 exple1 10342 leexp2rd 10447 facubnd 10484 permnn 10510 dfabsmax 10982 expcnvre 11265 dvdsadd2b 11529 dvdsmulgcd 11702 sqgcd 11706 bezoutr 11709 cncongr2 11774 pw2dvds 11833 hashgcdlem 11892 tgioo 12704 |
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