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Mirrors > Home > ILE Home > Th. List > rspec2 | GIF version |
Description: Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.) |
Ref | Expression |
---|---|
rspec2.1 | ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 |
Ref | Expression |
---|---|
rspec2 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspec2.1 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 | |
2 | 1 | rspec 2522 | . 2 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑) |
3 | 2 | r19.21bi 2558 | 1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2141 ∀wral 2448 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-4 1503 |
This theorem depends on definitions: df-bi 116 df-ral 2453 |
This theorem is referenced by: rspec3 2560 ordtriexmid 4503 onsucsssucexmid 4509 |
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