Theorem dvelimdc 2350

Description: Deduction form of dvelimc 2351. (Contributed by Mario Carneiro, 8-Oct-2016.)

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 1538	. . 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 2343	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑤 ∈ 𝐴)
6		dvelimdc.4	. . . . . 6 ⊢ (𝜑 → Ⅎ𝑧𝐵)
7	6	nfcrd 2343	. . . . 5 ⊢ (𝜑 → Ⅎ𝑧 𝑤 ∈ 𝐵)
8		dvelimdc.5	. . . . . 6 ⊢ (𝜑 → (𝑧 = 𝑦 → 𝐴 = 𝐵))
9		eleq2 2251	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵))
10	8, 9	syl6 33	. . . . 5 ⊢ (𝜑 → (𝑧 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵)))
11	2, 3, 5, 7, 10	dvelimdf 2026	. . . 4 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑤 ∈ 𝐵))
12	11	imp 124	. . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑤 ∈ 𝐵)
13	1, 12	nfcd 2324	. 2 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐵)
14	13	ex 115	1 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1361   = wceq 1363  Ⅎwnf 1470   ∈ wcel 2158  Ⅎwnfc 2316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-cleq 2180  df-clel 2183  df-nfc 2318

This theorem is referenced by:  dvelimc  2351