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Mirrors > Home > NFE 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 2772 | . 2 ⊢ (∃x ∈ A ∃y ∈ V φ ↔ ∃y ∈ V ∃x ∈ A φ) | |
2 | rexv 2873 | . . 3 ⊢ (∃y ∈ V φ ↔ ∃yφ) | |
3 | 2 | rexbii 2639 | . 2 ⊢ (∃x ∈ A ∃y ∈ V φ ↔ ∃x ∈ A ∃yφ) |
4 | rexv 2873 | . 2 ⊢ (∃y ∈ V ∃x ∈ A φ ↔ ∃y∃x ∈ A φ) | |
5 | 1, 3, 4 | 3bitr3i 266 | 1 ⊢ (∃x ∈ A ∃yφ ↔ ∃y∃x ∈ A φ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∃wex 1541 ∃wrex 2615 Vcvv 2859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-rex 2620 df-v 2861 |
This theorem is referenced by: rexcom4a 2879 reuind 3039 uni0b 3916 iuncom4 3976 dfiun2g 3999 iunn0 4026 iunxiun 4048 elpw12 4145 imacok 4282 unipw1 4325 dfaddc2 4381 addcass 4415 ltfinex 4464 ncfinlowerlem1 4482 nnpweqlem1 4522 vfinspss 4551 vfinncsp 4554 setconslem6 4736 xpiundi 4817 xpiundir 4818 cnvuni 4895 elimapw1 4944 elimapw12 4945 elimapw13 4946 elsnres 4996 imaco 5086 coiun 5090 fun11iun 5305 abrexco 5463 imaiun 5464 isomin 5496 dfdm4 5507 dfrn5 5508 xpassen 6057 enpw1pw 6075 |
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