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Theorem cbv3h 2441
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 8-Jun-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)
21nf5i 2191 . 2 𝑦𝜑
3 cbv3h.2 . . 3 (𝜓 → ∀𝑥𝜓)
43nf5i 2191 . 2 𝑥𝜓
5 cbv3h.3 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
62, 4, 5cbv3 2440 1 (∀𝑥𝜑 → ∀𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1860  df-nf 1864
This theorem is referenced by: (None)
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