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Theorem rexcom 2654
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 2332 . 2 𝑦𝐴
2 nfcv 2332 . 2 𝑥𝐵
31, 2rexcomf 2652 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 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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