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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 304 | . 2 ⊢ (𝜑 → (𝜏 ∧ 𝜂)) |
7 | syl122anc.6 | . 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂)) → 𝜁) | |
8 | 1, 2, 3, 6, 7 | syl121anc 1232 | 1 ⊢ (𝜑 → 𝜁) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 967 |
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 969 |
This theorem is referenced by: divdiv32apd 8703 divcanap5d 8704 divcanap7d 8706 divdivap1d 8709 divdivap2d 8710 seq3coll 10741 cau3lem 11042 summodclem2a 11308 prodmodclem2a 11503 prmind2 12031 divnumden 12107 pceulem 12205 pcqmul 12214 pcqdiv 12218 pcexp 12220 pcaddlem 12249 pcbc 12260 blss2ps 12953 blss2 12954 blssps 12974 blss 12975 xmeter 12983 |
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