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Theorem cbvaldva 1943
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 1542 . 2 𝑦𝜑
2 nfvd 1543 . 2 (𝜑 → Ⅎ𝑦𝜓)
3 cbvaldva.1 . . 3 ((𝜑𝑥 = 𝑦) → (𝜓𝜒))
43ex 115 . 2 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
51, 2, 4cbvald 1940 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549
This theorem depends on definitions:  df-bi 117  df-nf 1475
This theorem is referenced by:  cbvraldva2  2736
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