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Theorem cbvrmov 3426
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
cbvrmov.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 1917 . 2 𝑦𝜑
2 nfv 1917 . 2 𝑥𝜓
3 cbvrmov.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvrmo 3425 1 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  ∃*wrmo 3375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2371
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377
This theorem is referenced by: (None)
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