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Mirrors > Home > ILE Home > Th. List > rsp2 | GIF version |
Description: Restricted specialization. (Contributed by NM, 11-Feb-1997.) |
Ref | Expression |
---|---|
rsp2 | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rsp 2517 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑)) | |
2 | rsp 2517 | . . 3 ⊢ (∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → 𝜑)) | |
3 | 1, 2 | syl6 33 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑))) |
4 | 3 | impd 252 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ((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: ralidm 3514 sowlin 4303 cnmpt21 13006 cnmpt2t 13008 cnmpt22 13009 cnmptcom 13013 |
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