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Theorem cbvraldva 2637
Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
cbvraldva.1 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
Assertion
Ref Expression
cbvraldva (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑦𝐴 𝜒))
Distinct variable groups:   𝜓,𝑦   𝜒,𝑥   𝑥,𝐴,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)

Proof of Theorem cbvraldva
StepHypRef Expression
1 cbvraldva.1 . 2 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
2 eqidd 2118 . 2 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐴)
31, 2cbvraldva2 2635 1 (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑦𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wral 2393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-cleq 2110  df-clel 2113  df-ral 2398
This theorem is referenced by: (None)
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