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Theorem cbvaldva 2421
Description: Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. See also cbvaldvaw 2036. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
cbvaldva.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
Assertion
Ref Expression
cbvaldva (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Distinct variable groups:   𝜓,𝑦   𝜒,𝑥   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem cbvaldva
StepHypRef Expression
1 nfv 1906 . 2 𝑦𝜑
2 nfvd 1907 . 2 (𝜑 → Ⅎ𝑦𝜓)
3 cbvaldva.1 . . 3 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
43ex 413 . 2 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
51, 2, 4cbvald 2419 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-11 2151  ax-12 2167  ax-13 2381
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776
This theorem is referenced by:  cbval2vv  2426
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