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Mirrors > Home > MPE Home > Th. List > 2rmorex | Structured version Visualization version GIF version |
Description: Double restricted quantification with "at most one," analogous to 2moex 2531. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Ref | Expression |
---|---|
2rmorex | ⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2751 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
2 | nfre1 2988 | . . 3 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐵 𝜑 | |
3 | 1, 2 | nfrmo 3094 | . 2 ⊢ Ⅎ𝑦∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 |
4 | rspe 2986 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑦 ∈ 𝐵 𝜑) | |
5 | 4 | ex 449 | . . . . 5 ⊢ (𝑦 ∈ 𝐵 → (𝜑 → ∃𝑦 ∈ 𝐵 𝜑)) |
6 | 5 | ralrimivw 2950 | . . . 4 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 ∈ 𝐴 (𝜑 → ∃𝑦 ∈ 𝐵 𝜑)) |
7 | rmoim 3374 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ∃𝑦 ∈ 𝐵 𝜑) → (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝑦 ∈ 𝐵 → (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)) |
9 | 8 | com12 32 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → ∃*𝑥 ∈ 𝐴 𝜑)) |
10 | 3, 9 | ralrimi 2940 | 1 ⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ∃*wrmo 2899 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-eu 2462 df-mo 2463 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rmo 2904 |
This theorem is referenced by: 2reu2 39836 |
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