Theorem dvelimdc 2924

Description: Deduction form of dvelimc 2925. Usage of this theorem is discouraged because it depends on ax-13 2365. (Contributed by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dvelimdc.1	⊢ Ⅎ𝑥𝜑
dvelimdc.2	⊢ Ⅎ𝑧𝜑
dvelimdc.3	⊢ (𝜑 → Ⅎ𝑥𝐴)
dvelimdc.4	⊢ (𝜑 → Ⅎ𝑧𝐵)
dvelimdc.5	⊢ (𝜑 → (𝑧 = 𝑦 → 𝐴 = 𝐵))

Assertion

Ref	Expression
dvelimdc	⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵))

Proof of Theorem dvelimdc

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1909	. . 3 ⊢ Ⅎ𝑤(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
2		dvelimdc.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
3		dvelimdc.2	. . . . 5 ⊢ Ⅎ𝑧𝜑
4		dvelimdc.3	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5	4	nfcrd 2886	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑤 ∈ 𝐴)
6		dvelimdc.4	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑧𝐵)
7	6	nfcrd 2886	. . . . 5 ⊢ (𝜑 → Ⅎ𝑧 𝑤 ∈ 𝐵)
8		dvelimdc.5	. . . . . 6 ⊢ (𝜑 → (𝑧 = 𝑦 → 𝐴 = 𝐵))
9		eleq2 2816	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵))
10	8, 9	syl6 35	. . . . 5 ⊢ (𝜑 → (𝑧 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵)))
11	2, 3, 5, 7, 10	dvelimdf 2442	. . . 4 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑤 ∈ 𝐵))
12	11	imp 406	. . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑤 ∈ 𝐵)
13	1, 12	nfcd 2885	. 2 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐵)
14	13	ex 412	1 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1531   = wceq 1533  Ⅎwnf 1777   ∈ wcel 2098  Ⅎwnfc 2877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-13 2365  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-cleq 2718  df-clel 2804  df-nfc 2879

This theorem is referenced by:  dvelimc  2925