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Theorem dvelimdc 2340
Description: Deduction form of dvelimc 2341. (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 1528 . . 3 𝑤(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
2 dvelimdc.1 . . . . 5 𝑥𝜑
3 dvelimdc.2 . . . . 5 𝑧𝜑
4 dvelimdc.3 . . . . . 6 (𝜑𝑥𝐴)
54nfcrd 2333 . . . . 5 (𝜑 → Ⅎ𝑥 𝑤𝐴)
6 dvelimdc.4 . . . . . 6 (𝜑𝑧𝐵)
76nfcrd 2333 . . . . 5 (𝜑 → Ⅎ𝑧 𝑤𝐵)
8 dvelimdc.5 . . . . . 6 (𝜑 → (𝑧 = 𝑦𝐴 = 𝐵))
9 eleq2 2241 . . . . . 6 (𝐴 = 𝐵 → (𝑤𝐴𝑤𝐵))
108, 9syl6 33 . . . . 5 (𝜑 → (𝑧 = 𝑦 → (𝑤𝐴𝑤𝐵)))
112, 3, 5, 7, 10dvelimdf 2016 . . . 4 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑤𝐵))
1211imp 124 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑤𝐵)
131, 12nfcd 2314 . 2 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝐵)
1413ex 115 1 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wal 1351   = wceq 1353  wnf 1460  wcel 2148  wnfc 2306
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308
This theorem is referenced by:  dvelimc  2341
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