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Mirrors > Home > MPE Home > Th. List > 2reu2 | Structured version Visualization version GIF version |
Description: Double restricted existential uniqueness, analogous to 2eu2 2714. (Contributed by Alexander van der Vekens, 29-Jun-2017.) |
Ref | Expression |
---|---|
2reu2 | ⊢ (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 → (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reurmo 3378 | . . 3 ⊢ (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 → ∃*𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) | |
2 | 2rmorex 3693 | . . 3 ⊢ (∃*𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑) | |
3 | 2reu1 3826 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑))) | |
4 | simpl 486 | . . . 4 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) → ∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) | |
5 | 3, 4 | syl6bi 256 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 → ∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)) |
6 | 1, 2, 5 | 3syl 18 | . 2 ⊢ (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 → (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 → ∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)) |
7 | 2rexreu 3701 | . . 3 ⊢ ((∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) → ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑) | |
8 | 7 | expcom 417 | . 2 ⊢ (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 → (∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)) |
9 | 6, 8 | impbid 215 | 1 ⊢ (∃!𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 → (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wral 3106 ∃wrex 3107 ∃!wreu 3108 ∃*wrmo 3109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 |
This theorem is referenced by: 2reu8 43668 |
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