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Theorem rexcom 2598
Description: Commutation of restricted quantifiers. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
rexcom (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
Distinct variable groups:   𝑥,𝑦   𝑥,𝐵   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem rexcom
StepHypRef Expression
1 nfcv 2282 . 2 𝑦𝐴
2 nfcv 2282 . 2 𝑥𝐵
31, 2rexcomf 2596 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  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:  rexcom13  2599  rexcom4  2712  iuncom  3826  xpiundi  4604  addcomprg  7409  mulcomprg  7411  ltexprlemm  7431  caucvgprprlemexbt  7537  suplocexprlemml  7547  suplocexprlemmu  7549  qmulz  9441  elpq  9466  caubnd2  10920  sqrt2irr  11874
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