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Theorem dvelimdc 2509
 Description: Deduction form of dvelimc 2510. (Contributed by Mario Carneiro, 8-Oct-2016.)
Hypotheses
Ref Expression
dvelimdc.1 xφ
dvelimdc.2 zφ
dvelimdc.3 (φxA)
dvelimdc.4 (φzB)
dvelimdc.5 (φ → (z = yA = B))
Assertion
Ref Expression
dvelimdc (φ → (¬ x x = yxB))

Proof of Theorem dvelimdc
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . 3 w(φ ¬ x x = y)
2 dvelimdc.1 . . . . 5 xφ
3 dvelimdc.2 . . . . 5 zφ
4 dvelimdc.3 . . . . . 6 (φxA)
54nfcrd 2502 . . . . 5 (φ → Ⅎx w A)
6 dvelimdc.4 . . . . . 6 (φzB)
76nfcrd 2502 . . . . 5 (φ → Ⅎz w B)
8 dvelimdc.5 . . . . . 6 (φ → (z = yA = B))
9 eleq2 2414 . . . . . 6 (A = B → (w Aw B))
108, 9syl6 29 . . . . 5 (φ → (z = y → (w Aw B)))
112, 3, 5, 7, 10dvelimdf 2082 . . . 4 (φ → (¬ x x = y → Ⅎx w B))
1211imp 418 . . 3 ((φ ¬ x x = y) → Ⅎx w B)
131, 12nfcd 2484 . 2 ((φ ¬ x x = y) → xB)
1413ex 423 1 (φ → (¬ x x = yxB))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478 This theorem is referenced by:  dvelimc  2510
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