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Mirrors > Home > ILE Home > Th. List > alrimdd | GIF version |
Description: Deduction from Theorem 19.21 of [Margaris] p. 90. (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 | nfrd 1453 | . 2 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) |
3 | alrimdd.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
4 | alrimdd.3 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
5 | 3, 4 | alimd 1454 | . 2 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) |
6 | 2, 5 | syld 44 | 1 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1282 Ⅎwnf 1389 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-5 1376 ax-gen 1378 ax-4 1440 |
This theorem depends on definitions: df-bi 115 df-nf 1390 |
This theorem is referenced by: alrimd 1541 |
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