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Mirrors > Home > ILE Home > Th. List > rgen2 | GIF version |
Description: Generalization rule for restricted quantification. (Contributed by NM, 30-May-1999.) |
Ref | Expression |
---|---|
rgen2.1 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) |
Ref | Expression |
---|---|
rgen2 | ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rgen2.1 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) | |
2 | 1 | ralrimiva 2392 | . 2 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑) |
3 | 2 | rgen 2374 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1393 ∀wral 2306 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-4 1400 ax-17 1419 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-ral 2311 |
This theorem is referenced by: rgen3 2406 f1stres 5786 f2ndres 5787 divfnzn 8556 |
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