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Mirrors > Home > ILE Home > Th. List > rexcom4 | GIF version |
Description: Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Ref | Expression |
---|---|
rexcom4 | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom 2598 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ V 𝜑 ↔ ∃𝑦 ∈ V ∃𝑥 ∈ 𝐴 𝜑) | |
2 | rexv 2707 | . . 3 ⊢ (∃𝑦 ∈ V 𝜑 ↔ ∃𝑦𝜑) | |
3 | 2 | rexbii 2445 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ V 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦𝜑) |
4 | rexv 2707 | . 2 ⊢ (∃𝑦 ∈ V ∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑) | |
5 | 1, 3, 4 | 3bitr3i 209 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∃wex 1469 ∃wrex 2418 Vcvv 2689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rex 2423 df-v 2691 |
This theorem is referenced by: rexcom4a 2713 reuind 2893 iuncom4 3828 dfiun2g 3853 iunn0m 3881 iunxiun 3902 iinexgm 4087 inuni 4088 iunopab 4211 xpiundi 4605 xpiundir 4606 cnvuni 4733 dmiun 4756 elres 4863 elsnres 4864 rniun 4957 imaco 5052 coiun 5056 fun11iun 5396 abrexco 5668 imaiun 5669 fliftf 5708 rexrnmpo 5894 oprabrexex2 6036 releldm2 6091 eroveu 6528 genpassl 7356 genpassu 7357 ltexprlemopl 7433 ltexprlemopu 7435 ntreq0 12340 metrest 12714 |
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