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Theorem cbv3 1671
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 1509 . 2 𝑦𝑥𝜑
3 cbv3.2 . . 3 𝑥𝜓
4 cbv3.3 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4spim 1667 . 2 (∀𝑥𝜑𝜓)
62, 5alrimi 1456 1 (∀𝑥𝜑 → ∀𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283  wnf 1390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  cbv3h  1672  cbv1  1673  mo2n  1970  mo23  1983  setindis  10947  bdsetindis  10949
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