Theorem dvelimdc 2510

Description: Deduction form of dvelimc 2511. (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 1619	. . 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 2503	. . . . 5 ⊢ (φ → Ⅎx w ∈ A)
6		dvelimdc.4	. . . . . 6 ⊢ (φ → ℲzB)
7	6	nfcrd 2503	. . . . 5 ⊢ (φ → Ⅎz w ∈ B)
8		dvelimdc.5	. . . . . 6 ⊢ (φ → (z = y → A = B))
9		eleq2 2414	. . . . . 6 ⊢ (A = B → (w ∈ A ↔ w ∈ B))
10	8, 9	syl6 29	. . . . 5 ⊢ (φ → (z = y → (w ∈ A ↔ w ∈ B)))
11	2, 3, 5, 7, 10	dvelimdf 2082	. . . 4 ⊢ (φ → (¬ ∀x x = y → Ⅎx w ∈ B))
12	11	imp 418	. . 3 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎx w ∈ B)
13	1, 12	nfcd 2485	. 2 ⊢ ((φ ∧ ¬ ∀x x = y) → ℲxB)
14	13	ex 423	1 ⊢ (φ → (¬ ∀x x = y → ℲxB))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479

This theorem is referenced by:  dvelimc  2511