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Theorem cbvreuvw 2665
 Description: Version of cbvreuv 2661 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.)
Hypothesis
Ref Expression
cbvralvw.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvreuvw (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvreuvw
StepHypRef Expression
1 cbvralvw.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
21cbvreuv 2661 1 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∃!wreu 2420 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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2123 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1732  df-eu 1993  df-cleq 2134  df-clel 2137  df-reu 2425 This theorem is referenced by: (None)
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