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Theorem cbvrmov 3366
Description: Change the bound variable of a restricted at-most-one quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by Alexander van der Vekens, 17-Jun-2017.) (New usage is discouraged.)
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 1922 . 2 𝑦𝜑
2 nfv 1922 . 2 𝑥𝜓
3 cbvralv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvrmo 3357 1 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  ∃*wrmo 3064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-10 2141  ax-11 2158  ax-12 2175  ax-13 2371
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069
This theorem is referenced by: (None)
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