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Mirrors > Home > MPE Home > Th. List > Mathboxes > rexlim2d | Structured version Visualization version GIF version |
Description: Inference removing two restricted quantifiers. Same as rexlimdvv 3290, but with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
rexlim2d.x | ⊢ Ⅎ𝑥𝜑 |
rexlim2d.y | ⊢ Ⅎ𝑦𝜑 |
rexlim2d.3 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒))) |
Ref | Expression |
---|---|
rexlim2d | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexlim2d.x | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | nfv 1906 | . 2 ⊢ Ⅎ𝑥𝜒 | |
3 | rexlim2d.y | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
4 | nfv 1906 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 | |
5 | 3, 4 | nfan 1891 | . . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ 𝐴) |
6 | nfv 1906 | . . . 4 ⊢ Ⅎ𝑦𝜒 | |
7 | rexlim2d.3 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒))) | |
8 | 7 | expdimp 453 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 → (𝜓 → 𝜒))) |
9 | 5, 6, 8 | rexlimd 3314 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 𝜓 → 𝜒)) |
10 | 9 | ex 413 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝜓 → 𝜒))) |
11 | 1, 2, 10 | rexlimd 3314 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 Ⅎwnf 1775 ∈ wcel 2105 ∃wrex 3136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-ral 3140 df-rex 3141 |
This theorem is referenced by: fourierdlem48 42316 |
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