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Theorem bj-dvelimdv1 32810
Description: Curried (exported) form of bj-dvelimdv 32809. (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-dvelimdv.nf 𝑥𝜑
bj-dvelimdv.nf1 (𝜑 → Ⅎ𝑥𝜒)
bj-dvelimdv.is (𝑧 = 𝑦 → (𝜒𝜓))
Assertion
Ref Expression
bj-dvelimdv1 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧   𝜑,𝑧   𝜓,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦,𝑧)

Proof of Theorem bj-dvelimdv1
StepHypRef Expression
1 nfeqf2 2295 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
2 bj-dvelimdv.nf1 . . . 4 (𝜑 → Ⅎ𝑥𝜒)
3 nfimt 1819 . . . 4 (Ⅎ𝑥 𝑧 = 𝑦 → (Ⅎ𝑥𝜒 → Ⅎ𝑥(𝑧 = 𝑦𝜒)))
41, 2, 3syl2imc 41 . . 3 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑧 = 𝑦𝜒)))
54alrimdv 1855 . 2 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥(𝑧 = 𝑦𝜒)))
6 bj-nfalt 32677 . 2 (∀𝑧𝑥(𝑧 = 𝑦𝜒) → Ⅎ𝑥𝑧(𝑧 = 𝑦𝜒))
7 bj-dvelimdv.is . . . 4 (𝑧 = 𝑦 → (𝜒𝜓))
87equsalvw 1929 . . 3 (∀𝑧(𝑧 = 𝑦𝜒) ↔ 𝜓)
98nfbii 1776 . 2 (Ⅎ𝑥𝑧(𝑧 = 𝑦𝜒) ↔ Ⅎ𝑥𝜓)
105, 6, 9bj-syl66ib 32808 1 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1479  wnf 1706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1703  df-nf 1708
This theorem is referenced by:  bj-dvelimv  32811  bj-axc14nf  32813
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