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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 3293, 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 1911 | . 2 ⊢ Ⅎ𝑥𝜒 | |
3 | rexlim2d.y | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
4 | nfv 1911 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 | |
5 | 3, 4 | nfan 1896 | . . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ 𝐴) |
6 | nfv 1911 | . . . 4 ⊢ Ⅎ𝑦𝜒 | |
7 | rexlim2d.3 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒))) | |
8 | 7 | expdimp 455 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 → (𝜓 → 𝜒))) |
9 | 5, 6, 8 | rexlimd 3317 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 𝜓 → 𝜒)) |
10 | 9 | ex 415 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝜓 → 𝜒))) |
11 | 1, 2, 10 | rexlimd 3317 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 Ⅎwnf 1780 ∈ wcel 2110 ∃wrex 3139 |
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 1907 ax-6 1966 ax-7 2011 ax-12 2172 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-ral 3143 df-rex 3144 |
This theorem is referenced by: fourierdlem48 42432 |
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