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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 2235 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2235 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | rexcomf 2543 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∃wrex 2371 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-nf 1402 df-sb 1700 df-cleq 2088 df-clel 2091 df-nfc 2224 df-rex 2376 |
This theorem is referenced by: rexcom13 2546 rexcom4 2656 iuncom 3758 xpiundi 4525 addcomprg 7234 mulcomprg 7236 ltexprlemm 7256 caucvgprprlemexbt 7362 qmulz 9207 caubnd2 10681 sqrt2irr 11583 |
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