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Mirrors > Home > MPE Home > Th. List > rsp2e | Structured version Visualization version GIF version |
Description: Restricted specialization. (Contributed by FL, 4-Jun-2012.) (Proof shortened by Wolf Lammen, 7-Jan-2020.) |
Ref | Expression |
---|---|
rsp2e | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspe 3304 | . . 3 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑦 ∈ 𝐵 𝜑) | |
2 | rspe 3304 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) | |
3 | 1, 2 | sylan2 594 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) |
4 | 3 | 3impb 1111 | 1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 ∃wrex 3139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-3an 1085 df-ex 1781 df-rex 3144 |
This theorem is referenced by: pell14qrdich 39486 |
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