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Theorem cbvreuv 3357
Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. See cbvreuvw 3353 for a version without ax-13 2373, but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2373. Use the weaker cbvreuvw 3353 when possible. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
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 1921 . 2 𝑦𝜑
2 nfv 1921 . 2 𝑥𝜓
3 cbvralv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbvreu 3348 1 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  ∃!wreu 3056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-10 2145  ax-11 2162  ax-12 2179  ax-13 2373
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clel 2812  df-reu 3061
This theorem is referenced by: (None)
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