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Theorem cbv2 2375
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 2227 . . 3 (𝜑 → ∀𝑦𝜑)
41, 3alrimi 2246 . 2 (𝜑 → ∀𝑥𝑦𝜑)
5 cbv2.3 . . . 4 (𝜑 → Ⅎ𝑦𝜓)
65nf5rd 2228 . . 3 (𝜑 → (𝜓 → ∀𝑦𝜓))
7 cbv2.4 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
87nf5rd 2228 . . 3 (𝜑 → (𝜒 → ∀𝑥𝜒))
9 cbv2.5 . . 3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
106, 8, 9cbv2h 2374 . 2 (∀𝑥𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
114, 10syl 17 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1650  wnf 1878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-ex 1875  df-nf 1879
This theorem is referenced by:  cbvald  2380  sb9  2517  wl-cbvalnaed  33765  wl-sb8t  33780
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