Theorem bj-dvelimdv1 33354

Description: Curried (exported) form of bj-dvelimdv 33353 (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 → (𝜒 ↔ 𝜓))

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

Step	Hyp	Ref	Expression
1		nfeqf2 2394	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑧 = 𝑦)
2		bj-dvelimdv.nf	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒)
3		bj-nfimt 33141	. . . 4 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 → (Ⅎ𝑥𝜒 → Ⅎ𝑥(𝑧 = 𝑦 → 𝜒)))
4	1, 2, 3	syl2imc 41	. . 3 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑧 = 𝑦 → 𝜒)))
5	4	alrimdv 2028	. 2 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧Ⅎ𝑥(𝑧 = 𝑦 → 𝜒)))
6		bj-nfalt 33235	. 2 ⊢ (∀𝑧Ⅎ𝑥(𝑧 = 𝑦 → 𝜒) → Ⅎ𝑥∀𝑧(𝑧 = 𝑦 → 𝜒))
7		bj-dvelimdv.is	. . . 4 ⊢ (𝑧 = 𝑦 → (𝜒 ↔ 𝜓))
8	7	equsalvw 2108	. . 3 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜒) ↔ 𝜓)
9	8	nfbii 1951	. 2 ⊢ (Ⅎ𝑥∀𝑧(𝑧 = 𝑦 → 𝜒) ↔ Ⅎ𝑥𝜓)
10	5, 6, 9	bj-syl66ib 33066	1 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198  ∀wal 1654  Ⅎwnf 1882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ex 1879  df-nf 1883

This theorem is referenced by:  bj-dvelimv  33355  bj-axc14nf  33357