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Mirrors > Home > MPE Home > Th. List > syl1111anc | Structured version Visualization version GIF version |
Description: Four-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl112anc 1374 except the unification theorem uses left-nested conjunction. (Contributed by Alan Sare, 17-Oct-2017.) |
Ref | Expression |
---|---|
syl1111anc.1 | ⊢ (𝜑 → 𝜓) |
syl1111anc.2 | ⊢ (𝜑 → 𝜒) |
syl1111anc.3 | ⊢ (𝜑 → 𝜃) |
syl1111anc.4 | ⊢ (𝜑 → 𝜏) |
syl1111anc.5 | ⊢ ((((𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂) |
Ref | Expression |
---|---|
syl1111anc | ⊢ (𝜑 → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl1111anc.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | syl1111anc.2 | . . 3 ⊢ (𝜑 → 𝜒) | |
3 | 1, 2 | jca 511 | . 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒)) |
4 | syl1111anc.3 | . 2 ⊢ (𝜑 → 𝜃) | |
5 | syl1111anc.4 | . 2 ⊢ (𝜑 → 𝜏) | |
6 | syl1111anc.5 | . 2 ⊢ ((((𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂) | |
7 | 3, 4, 5, 6 | syl21anc 837 | 1 ⊢ (𝜑 → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 |
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 207 df-an 396 |
This theorem is referenced by: mpsyl4anc 841 ucnima 24311 f1otrge 28898 swrdf1 32923 mgcf1o 32976 chnind 32983 cycpmrn 33136 linds2eq 33374 rhmimaidl 33425 idlmulssprm 33435 isprmidlc 33440 prmidlc 33441 qsidomlem2 33446 ply1unit 33565 lbsdiflsp0 33639 extdg1id 33676 3cubeslem1 42640 cantnftermord 43282 sineq0ALT 44908 cncfshift 45795 cncfperiod 45800 |
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