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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 2543. (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 2764 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
2 | nfre1 3005 | . . 3 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐵 𝜑 | |
3 | 1, 2 | nfrmo 3115 | . 2 ⊢ Ⅎ𝑦∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 |
4 | rmoim 3407 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ∃𝑦 ∈ 𝐵 𝜑) → (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 𝜑)) | |
5 | rspe 3003 | . . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑦 ∈ 𝐵 𝜑) | |
6 | 5 | ex 450 | . . . 4 ⊢ (𝑦 ∈ 𝐵 → (𝜑 → ∃𝑦 ∈ 𝐵 𝜑)) |
7 | 6 | ralrimivw 2967 | . . 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 2957 | 1 ⊢ (∃*𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∀𝑦 ∈ 𝐵 ∃*𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ∃*wrmo 2915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-eu 2474 df-mo 2475 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rmo 2920 |
This theorem is referenced by: 2reu2 41187 |
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