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Theorem bj-dvelimdv1 34252
 Description: Curried (exported) form of bj-dvelimdv 34251 (of course, one is directly provable from the other, but we keep this proof for illustration purposes). (Contributed by BJ, 20-Oct-2021.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-dvelimdv.nf (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 2396 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
2 bj-dvelimdv.nf . . . 4 (𝜑 → Ⅎ𝑥𝜒)
3 bj-nfimt 34045 . . . 4 (Ⅎ𝑥 𝑧 = 𝑦 → (Ⅎ𝑥𝜒 → Ⅎ𝑥(𝑧 = 𝑦𝜒)))
41, 2, 3syl2imc 41 . . 3 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑧 = 𝑦𝜒)))
54alrimdv 1930 . 2 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥(𝑧 = 𝑦𝜒)))
6 bj-nfalt 34119 . 2 (∀𝑧𝑥(𝑧 = 𝑦𝜒) → Ⅎ𝑥𝑧(𝑧 = 𝑦𝜒))
7 bj-dvelimdv.is . . . 4 (𝑧 = 𝑦 → (𝜒𝜓))
87equsalvw 2010 . . 3 (∀𝑧(𝑧 = 𝑦𝜒) ↔ 𝜓)
98nfbii 1853 . 2 (Ⅎ𝑥𝑧(𝑧 = 𝑦𝜒) ↔ Ⅎ𝑥𝜓)
105, 6, 9bj-syl66ib 33964 1 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1536  Ⅎwnf 1785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2178  ax-13 2391 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786 This theorem is referenced by:  bj-dvelimv  34253  bj-axc14nf  34255
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