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Theorem cbvalv 2420
Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2392. See cbvalvw 2044 for a version requiring fewer axioms, to be preferred when sufficient. (Contributed by NM, 5-Aug-1993.) Remove dependency on ax-10 2146, shorten. (Revised by Wolf Lammen, 11-Sep-2023.) (New usage is discouraged.)
Hypothesis
Ref Expression
cbvalv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvalv (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvalv
StepHypRef Expression
1 nfv 1916 . 2 𝑦𝜑
2 nfv 1916 . 2 𝑥𝜓
3 cbvalv.1 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
41, 2, 3cbval 2418 1 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536
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 1912  ax-6 1971  ax-7 2016  ax-11 2162  ax-12 2179  ax-13 2392
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786
This theorem is referenced by:  cbvexvOLD  2423  cbval2vv  2437  cdeqal1  3748
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