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Mirrors > Home > ILE Home > Th. List > al2imi | GIF version |
Description: Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
al2imi.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
al2imi | ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | al2imi.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | 1 | alimi 1432 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜒)) |
3 | alim 1434 | . 2 ⊢ (∀𝑥(𝜓 → 𝜒) → (∀𝑥𝜓 → ∀𝑥𝜒)) | |
4 | 2, 3 | syl 14 | 1 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-5 1424 ax-gen 1426 |
This theorem is referenced by: alanimi 1436 alimdh 1444 albi 1445 19.30dc 1607 19.33b2 1609 hbnt 1632 ax10o 1694 spimth 1714 sbi1v 1864 ralim 2494 ceqsalt 2715 intss 3800 |
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