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Theorem cbvcllem 37423
 Description: Change of bound variable in class of supersets of a with a property. (Contributed by RP, 24-Jul-2020.)
Hypothesis
Ref Expression
cbvcllem.y (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvcllem {𝑥 ∣ (𝑋𝑥𝜑)} = {𝑦 ∣ (𝑋𝑦𝜓)}
Distinct variable groups:   𝑥,𝑦,𝑋   𝜓,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvcllem
StepHypRef Expression
1 cbvcllem.y . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
21cleq2lem 37422 . 2 (𝑥 = 𝑦 → ((𝑋𝑥𝜑) ↔ (𝑋𝑦𝜓)))
32cbvabv 2744 1 {𝑥 ∣ (𝑋𝑥𝜑)} = {𝑦 ∣ (𝑋𝑦𝜓)}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480  {cab 2607   ⊆ wss 3559 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-in 3566  df-ss 3573 This theorem is referenced by: (None)
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