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Mirrors > Home > MPE Home > Th. List > rsp2 | Structured version Visualization version GIF version |
Description: Restricted specialization, with two quantifiers. (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 3245 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑)) | |
2 | rsp 3245 | . . 3 ⊢ (∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → 𝜑)) | |
3 | 1, 2 | syl6 35 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑))) |
4 | 3 | impd 410 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ∀wral 3059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-ral 3060 |
This theorem is referenced by: ralcom2 3375 disjxiun 5145 mpocurryd 8293 cmncom 19831 cnmpt21 23695 cnmpt2t 23697 cnmpt22 23698 cnmptcom 23702 frgrwopreglem5ALT 30351 htthlem 30946 qsidomlem2 33461 cplgredgex 35105 disjlem14 38780 prtlem14 38856 islptre 45575 sprsymrelfolem2 47418 |
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