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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 2332 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2332 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | rexcomf 2652 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∃wrex 2469 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rex 2474 |
This theorem is referenced by: rexcom13 2656 rexcom4 2775 iuncom 3910 xpiundi 4705 addcomprg 7612 mulcomprg 7614 ltexprlemm 7634 caucvgprprlemexbt 7740 suplocexprlemml 7750 suplocexprlemmu 7752 qmulz 9659 elpq 9684 caubnd2 11167 sqrt2irr 12205 pythagtriplem19 12325 |
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