Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 1366 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 512 | . 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 833 | 1 ⊢ (𝜑 → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 |
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 208 df-an 397 |
This theorem is referenced by: mpsyl4anc 836 ucnima 22817 f1otrge 26585 swrdf1 30557 cycpmrn 30712 linds2eq 30868 isprmidlc 30881 qsidomlem2 30883 lbsdiflsp0 30921 extdg1id 30952 3cubeslem1 39159 sineq0ALT 41148 cncfshift 42033 cncfperiod 42038 |
Copyright terms: Public domain | W3C validator |