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Theorem cbvaldva 1878
Description: Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. (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 1491 . 2 𝑦𝜑
2 nfvd 1492 . 2 (𝜑 → Ⅎ𝑦𝜓)
3 cbvaldva.1 . . 3 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
43ex 114 . 2 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
51, 2, 4cbvald 1875 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1312
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-nf 1420
This theorem is referenced by:  cbvraldva2  2632
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