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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 2306 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2306 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | rexcomf 2626 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∃wrex 2443 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 |
This theorem is referenced by: rexcom13 2629 rexcom4 2744 iuncom 3866 xpiundi 4656 addcomprg 7510 mulcomprg 7512 ltexprlemm 7532 caucvgprprlemexbt 7638 suplocexprlemml 7648 suplocexprlemmu 7650 qmulz 9552 elpq 9577 caubnd2 11045 sqrt2irr 12071 pythagtriplem19 12191 |
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