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Theorem cbvreuv 2614
Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypothesis
Ref Expression
cbvralv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvreuv (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvreuv
StepHypRef Expression
1 nfv 1476 . 2 𝑦𝜑
2 nfv 1476 . 2 𝑥𝜓
3 cbvralv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvreu 2610 1 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  ∃!wreu 2377
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 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-cleq 2093  df-clel 2096  df-reu 2382
This theorem is referenced by:  reu8  2833
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