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Theorem cbv3 1721
 Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-May-2018.)
Hypotheses
Ref Expression
cbv3.1 𝑦𝜑
cbv3.2 𝑥𝜓
cbv3.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbv3 (∀𝑥𝜑 → ∀𝑦𝜓)

Proof of Theorem cbv3
StepHypRef Expression
1 cbv3.1 . . 3 𝑦𝜑
21nfal 1556 . 2 𝑦𝑥𝜑
3 cbv3.2 . . 3 𝑥𝜓
4 cbv3.3 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4spim 1717 . 2 (∀𝑥𝜑𝜓)
62, 5alrimi 1503 1 (∀𝑥𝜑 → ∀𝑦𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1330  Ⅎwnf 1437 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-i9 1511  ax-ial 1515 This theorem depends on definitions:  df-bi 116  df-nf 1438 This theorem is referenced by:  cbv3h  1722  cbv1  1723  mo2n  2028  mo23  2041  setindis  13356  bdsetindis  13358
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