![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > alrimdd | Structured version Visualization version GIF version |
Description: Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2206. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Ref | Expression |
---|---|
alrimdd.1 | ⊢ Ⅎ𝑥𝜑 |
alrimdd.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
alrimdd.3 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
alrimdd | ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alrimdd.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
2 | 1 | nf5rd 2195 | . 2 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) |
3 | alrimdd.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
4 | alrimdd.3 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
5 | 3, 4 | alimd 2211 | . 2 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) |
6 | 2, 5 | syld 47 | 1 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1537 Ⅎwnf 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-12 2176 |
This theorem depends on definitions: df-bi 210 df-ex 1783 df-nf 1787 |
This theorem is referenced by: alrimd 2214 wl-euequf 35240 |
Copyright terms: Public domain | W3C validator |