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Theorem cbvald 1875
Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 1968. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)
Hypotheses
Ref Expression
cbvald.1 𝑦𝜑
cbvald.2 (𝜑 → Ⅎ𝑦𝜓)
cbvald.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbvald (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Distinct variable groups:   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑦)

Proof of Theorem cbvald
StepHypRef Expression
1 nfv 1491 . 2 𝑥𝜑
2 cbvald.1 . 2 𝑦𝜑
3 cbvald.2 . 2 (𝜑 → Ⅎ𝑦𝜓)
4 nfv 1491 . . 3 𝑥𝜒
54a1i 9 . 2 (𝜑 → Ⅎ𝑥𝜒)
6 cbvald.3 . 2 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
71, 2, 3, 5, 6cbv2 1708 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1312  wnf 1419
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 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:  cbvaldva  1878
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