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Theorem cbv2 2306
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) Format hypotheses to common style. (Revised by Wolf Lammen, 13-May-2018.)
Hypotheses
Ref Expression
cbv2.1 𝑥𝜑
cbv2.2 𝑦𝜑
cbv2.3 (𝜑 → Ⅎ𝑦𝜓)
cbv2.4 (𝜑 → Ⅎ𝑥𝜒)
cbv2.5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbv2 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Proof of Theorem cbv2
StepHypRef Expression
1 cbv2.1 . . 3 𝑥𝜑
2 cbv2.2 . . . 4 𝑦𝜑
32nf5ri 2103 . . 3 (𝜑 → ∀𝑦𝜑)
41, 3alrimi 2120 . 2 (𝜑 → ∀𝑥𝑦𝜑)
5 cbv2.3 . . . 4 (𝜑 → Ⅎ𝑦𝜓)
65nf5rd 2104 . . 3 (𝜑 → (𝜓 → ∀𝑦𝜓))
7 cbv2.4 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
87nf5rd 2104 . . 3 (𝜑 → (𝜒 → ∀𝑥𝜒))
9 cbv2.5 . . 3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
106, 8, 9cbv2h 2305 . 2 (∀𝑥𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
114, 10syl 17 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1521  wnf 1748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750
This theorem is referenced by:  cbvald  2313  sb9  2454  wl-cbvalnaed  33449  wl-sb8t  33463
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