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Mirrors > Home > MPE Home > Th. List > rexun | Structured version Visualization version GIF version |
Description: Restricted existential quantification over union. (Contributed by Jeff Madsen, 5-Jan-2011.) |
Ref | Expression |
---|---|
rexun | ⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐵 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2918 | . 2 ⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ ∃𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑)) | |
2 | 19.43 1810 | . . 3 ⊢ (∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ 𝜑)) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∨ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))) | |
3 | elun 3753 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) | |
4 | 3 | anbi1i 731 | . . . . 5 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝜑)) |
5 | andir 912 | . . . . 5 ⊢ (((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ 𝜑))) | |
6 | 4, 5 | bitri 264 | . . . 4 ⊢ ((𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ 𝜑))) |
7 | 6 | exbii 1774 | . . 3 ⊢ (∃𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑) ↔ ∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) ∨ (𝑥 ∈ 𝐵 ∧ 𝜑))) |
8 | df-rex 2918 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
9 | df-rex 2918 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) | |
10 | 8, 9 | orbi12i 543 | . . 3 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐵 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∨ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))) |
11 | 2, 7, 10 | 3bitr4i 292 | . 2 ⊢ (∃𝑥(𝑥 ∈ (𝐴 ∪ 𝐵) ∧ 𝜑) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐵 𝜑)) |
12 | 1, 11 | bitri 264 | 1 ⊢ (∃𝑥 ∈ (𝐴 ∪ 𝐵)𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∨ ∃𝑥 ∈ 𝐵 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∨ wo 383 ∧ wa 384 ∃wex 1704 ∈ wcel 1990 ∃wrex 2913 ∪ cun 3572 |
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-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-v 3202 df-un 3579 |
This theorem is referenced by: rexprg 4235 rextpg 4237 iunxun 4605 oarec 7642 zornn0g 9327 scshwfzeqfzo 13572 rpnnen2lem12 14954 dvdsprmpweqnn 15589 vdwlem6 15690 pmatcollpw3fi1 20593 cmpfi 21211 poimirlem25 33434 unima 39346 |
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