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Theorem cbv3 2411
Description: Rule used to change bound variables, using implicit substitution, that does not use ax-c9 35912. (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 . . . 4 𝑦𝜑
21nf5ri 2187 . . 3 (𝜑 → ∀𝑦𝜑)
32hbal 2166 . 2 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
4 cbv3.2 . . 3 𝑥𝜓
5 cbv3.3 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
64, 5spim 2401 . 2 (∀𝑥𝜑𝜓)
73, 6alrimih 1817 1 (∀𝑥𝜑 → ∀𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1528  wnf 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-11 2153  ax-12 2169  ax-13 2385
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-nf 1778
This theorem is referenced by:  cbval  2412  cbv1  2418  cbv3h  2420  axc16i  2455
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