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Theorem cbvrmov 2693
Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Hypothesis
Ref Expression
cbvralv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvrmov (∃*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 cbvrmov
StepHypRef Expression
1 nfv 1515 . 2 𝑦𝜑
2 nfv 1515 . 2 𝑥𝜓
3 cbvralv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvrmo 2689 1 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  ∃*wrmo 2445
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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rex 2448  df-reu 2449  df-rmo 2450
This theorem is referenced by: (None)
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