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Theorem cbv3h 1723
 Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-May-2018.)
Hypotheses
Ref Expression
cbv3h.1 (𝜑 → ∀𝑦𝜑)
cbv3h.2 (𝜓 → ∀𝑥𝜓)
cbv3h.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbv3h (∀𝑥𝜑 → ∀𝑦𝜓)

Proof of Theorem cbv3h
StepHypRef Expression
1 cbv3h.1 . . 3 (𝜑 → ∀𝑦𝜑)
21nfi 1442 . 2 𝑦𝜑
3 cbv3h.2 . . 3 (𝜓 → ∀𝑥𝜓)
43nfi 1442 . 2 𝑥𝜓
5 cbv3h.3 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
62, 4, 5cbv3 1722 1 (∀𝑥𝜑 → ∀𝑦𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1333 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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-i9 1510  ax-ial 1514 This theorem depends on definitions:  df-bi 116  df-nf 1441 This theorem is referenced by:  cbvalh  1733  ax16  1793  ax16i  1838  cleqh  2257
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