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Mirrors > Home > ILE Home > Th. List > reeanv | GIF version |
Description: Rearrange existential quantifiers. (Contributed by NM, 9-May-1999.) |
Ref | Expression |
---|---|
reeanv | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1509 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1509 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | 1, 2 | reean 2602 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∃wrex 2418 |
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-nf 1438 df-sb 1737 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rex 2423 |
This theorem is referenced by: 3reeanv 2604 fliftfun 5705 tfrlem5 6219 eroveu 6528 erovlem 6529 xpf1o 6746 genprndl 7353 genprndu 7354 ltpopr 7427 ltsopr 7428 cauappcvgprlemdisj 7483 caucvgprlemdisj 7506 caucvgprprlemdisj 7534 exbtwnzlemex 10058 rebtwn2z 10063 rexanre 11024 summodc 11184 prodmodclem2 11378 prodmodc 11379 dvds2lem 11541 odd2np1 11606 opoe 11628 omoe 11629 opeo 11630 omeo 11631 gcddiv 11743 divgcdcoprmex 11819 tgcl 12272 restbasg 12376 txuni2 12464 txbas 12466 txcnp 12479 blin2 12640 tgqioo 12755 |
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