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Mirrors > Home > ILE Home > Th. List > syl121anc | 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) |
syl121anc.5 | ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂) |
Ref | Expression |
---|---|
syl121anc | ⊢ (𝜑 → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sylXanc.1 | . 2 ⊢ (𝜑 → 𝜓) | |
2 | sylXanc.2 | . . 3 ⊢ (𝜑 → 𝜒) | |
3 | sylXanc.3 | . . 3 ⊢ (𝜑 → 𝜃) | |
4 | 2, 3 | jca 304 | . 2 ⊢ (𝜑 → (𝜒 ∧ 𝜃)) |
5 | sylXanc.4 | . 2 ⊢ (𝜑 → 𝜏) | |
6 | syl121anc.5 | . 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂) | |
7 | 1, 4, 5, 6 | syl3anc 1216 | 1 ⊢ (𝜑 → 𝜂) |
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: syl122anc 1225 tfisi 4501 tfrcllemsucfn 6250 sbthlemi6 6850 sbthlemi8 6852 div32apd 8574 div13apd 8575 expdivapd 10438 modfsummodlemstep 11226 ennnfonelemg 11916 ssblex 12600 |
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