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Theorem cbv3 2404
Description: Rule used to change bound variables, using implicit substitution, that does not use ax-c9 34911. (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 2346 . 2 𝑦𝑥𝜑
3 cbv3.2 . . 3 𝑥𝜓
4 cbv3.3 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4spim 2394 . 2 (∀𝑥𝜑𝜓)
62, 5alrimi 2248 1 (∀𝑥𝜑 → ∀𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1651  wnf 1879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377
This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-nf 1880
This theorem is referenced by:  cbv3h  2405  cbv1  2406  cbval  2410  axc16i  2443  bj-mo3OLD  33327
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