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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 2913 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑)) | |
2 | rsp 2913 | . . 3 ⊢ (∀𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → 𝜑)) | |
3 | 1, 2 | syl6 34 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝜑))) |
4 | 3 | impd 446 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-ral 2901 |
This theorem is referenced by: ralcom2 3083 disjxiun 4579 disjxiunOLD 4580 solin 4982 mpt2curryd 7282 cmncom 18032 cnmpt21 21284 cnmpt2t 21286 cnmpt22 21287 cnmptcom 21291 frgrawopreglem5 26575 htthlem 27158 prtlem14 33177 islptre 38686 frgrwopreglem5 41485 |
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