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Theorem cbv1h 2429
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 11-May-1993.) (Proof shortened by Wolf Lammen, 13-May-2018.)
Hypotheses
Ref Expression
cbv1h.1 (𝜑 → (𝜓 → ∀𝑦𝜓))
cbv1h.2 (𝜑 → (𝜒 → ∀𝑥𝜒))
cbv1h.3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbv1h (∀𝑥𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Proof of Theorem cbv1h
StepHypRef Expression
1 nfa1 2184 . 2 𝑥𝑥𝑦𝜑
2 nfa2 2196 . 2 𝑦𝑥𝑦𝜑
3 2sp 2210 . . . 4 (∀𝑥𝑦𝜑𝜑)
4 cbv1h.1 . . . 4 (𝜑 → (𝜓 → ∀𝑦𝜓))
53, 4syl 17 . . 3 (∀𝑥𝑦𝜑 → (𝜓 → ∀𝑦𝜓))
62, 5nf5d 2281 . 2 (∀𝑥𝑦𝜑 → Ⅎ𝑦𝜓)
7 cbv1h.2 . . . 4 (𝜑 → (𝜒 → ∀𝑥𝜒))
83, 7syl 17 . . 3 (∀𝑥𝑦𝜑 → (𝜒 → ∀𝑥𝜒))
91, 8nf5d 2281 . 2 (∀𝑥𝑦𝜑 → Ⅎ𝑥𝜒)
10 cbv1h.3 . . 3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
113, 10syl 17 . 2 (∀𝑥𝑦𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
121, 2, 6, 9, 11cbv1 2428 1 (∀𝑥𝑦𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ex 1853  df-nf 1858
This theorem is referenced by:  cbv2h  2430
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