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Mirrors > Home > ILE Home > Th. List > rexcom | GIF version |
Description: Commutation of restricted quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
rexcom | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2319 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2319 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | rexcomf 2639 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∃wrex 2456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 |
This theorem is referenced by: rexcom13 2643 rexcom4 2761 iuncom 3893 xpiundi 4685 addcomprg 7577 mulcomprg 7579 ltexprlemm 7599 caucvgprprlemexbt 7705 suplocexprlemml 7715 suplocexprlemmu 7717 qmulz 9623 elpq 9648 caubnd2 11126 sqrt2irr 12162 pythagtriplem19 12282 |
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