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Theorem cbv2OLD 2421
Description: Obsolete version of cbv2 2417 as of 10-Sep-2023. (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.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
cbv2OLD.1 𝑥𝜑
cbv2OLD.2 𝑦𝜑
cbv2OLD.3 (𝜑 → Ⅎ𝑦𝜓)
cbv2OLD.4 (𝜑 → Ⅎ𝑥𝜒)
cbv2OLD.5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbv2OLD (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Proof of Theorem cbv2OLD
StepHypRef Expression
1 cbv2OLD.1 . . 3 𝑥𝜑
2 cbv2OLD.2 . . . 4 𝑦𝜑
32nf5ri 2188 . . 3 (𝜑 → ∀𝑦𝜑)
41, 3alrimi 2206 . 2 (𝜑 → ∀𝑥𝑦𝜑)
5 cbv2OLD.3 . . . 4 (𝜑 → Ⅎ𝑦𝜓)
65nf5rd 2189 . . 3 (𝜑 → (𝜓 → ∀𝑦𝜓))
7 cbv2OLD.4 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
87nf5rd 2189 . . 3 (𝜑 → (𝜒 → ∀𝑥𝜒))
9 cbv2OLD.5 . . 3 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
106, 8, 9cbv2h 2420 . 2 (∀𝑥𝑦𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
114, 10syl 17 1 (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wal 1529  wnf 1778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-10 2139  ax-11 2154  ax-12 2170  ax-13 2384
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1775  df-nf 1779
This theorem is referenced by: (None)
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