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Mirrors > Home > ILE Home > Th. List > ral2imi | GIF version |
Description: Inference quantifying antecedent, nested antecedent, and consequent, with a strong hypothesis. (Contributed by NM, 19-Dec-2006.) |
Ref | Expression |
---|---|
ral2imi.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
ral2imi | ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ral2imi.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | 1 | ralimi 2467 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) |
3 | ralim 2463 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) | |
4 | 2, 3 | syl 14 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wral 2388 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1404 ax-gen 1406 |
This theorem depends on definitions: df-bi 116 df-ral 2393 |
This theorem is referenced by: r19.26 2530 iinerm 6453 ss2ixp 6557 bj-findis 12860 |
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