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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 2640. (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 2905 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
| 2 | nfre1 3285 | . . 3 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐵 𝜑 | |
| 3 | 1, 2 | nfrmow 3413 | . 2 ⊢ Ⅎ𝑦∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 | 
| 4 | rmoim 3746 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ∃𝑦 ∈ 𝐵 𝜑) → (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)) | |
| 5 | rspe 3249 | . . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑦 ∈ 𝐵 𝜑) | |
| 6 | 5 | ex 412 | . . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝜑 → ∃𝑦 ∈ 𝐵 𝜑)) | 
| 7 | 6 | ralrimivw 3150 | . . 3 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 ∈ 𝐴 (𝜑 → ∃𝑦 ∈ 𝐵 𝜑)) | 
| 8 | 4, 7 | syl11 33 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → (𝑦 ∈ 𝐵 → ∃*𝑥 ∈ 𝐴 𝜑)) | 
| 9 | 3, 8 | ralrimi 3257 | 1 ⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ∃*wrmo 3379 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-10 2141 ax-11 2157 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-mo 2540 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rmo 3380 | 
| This theorem is referenced by: 2reu2 3898 | 
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