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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 2637. (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 2904 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
2 | nfre1 3283 | . . 3 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐵 𝜑 | |
3 | 1, 2 | nfrmow 3410 | . 2 ⊢ Ⅎ𝑦∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 |
4 | rmoim 3737 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ∃𝑦 ∈ 𝐵 𝜑) → (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)) | |
5 | rspe 3247 | . . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑦 ∈ 𝐵 𝜑) | |
6 | 5 | ex 414 | . . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝜑 → ∃𝑦 ∈ 𝐵 𝜑)) |
7 | 6 | ralrimivw 3151 | . . 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 3255 | 1 ⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ∃*wrmo 3376 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-10 2138 ax-11 2155 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-mo 2535 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rmo 3377 |
This theorem is referenced by: 2reu2 3893 |
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