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Theorem dvelimdc 2771
Description: Deduction form of dvelimc 2772. (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 1829 . . 3 𝑤(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
2 dvelimdc.1 . . . . 5 𝑥𝜑
3 dvelimdc.2 . . . . 5 𝑧𝜑
4 dvelimdc.3 . . . . . 6 (𝜑𝑥𝐴)
54nfcrd 2756 . . . . 5 (𝜑 → Ⅎ𝑥 𝑤𝐴)
6 dvelimdc.4 . . . . . 6 (𝜑𝑧𝐵)
76nfcrd 2756 . . . . 5 (𝜑 → Ⅎ𝑧 𝑤𝐵)
8 dvelimdc.5 . . . . . 6 (𝜑 → (𝑧 = 𝑦𝐴 = 𝐵))
9 eleq2 2676 . . . . . 6 (𝐴 = 𝐵 → (𝑤𝐴𝑤𝐵))
108, 9syl6 34 . . . . 5 (𝜑 → (𝑧 = 𝑦 → (𝑤𝐴𝑤𝐵)))
112, 3, 5, 7, 10dvelimdf 2322 . . . 4 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑤𝐵))
1211imp 443 . . 3 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑤𝐵)
131, 12nfcd 2745 . 2 ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → 𝑥𝐵)
1413ex 448 1 (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  wal 1472   = wceq 1474  wnf 1698  wcel 1976  wnfc 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-cleq 2602  df-clel 2605  df-nfc 2739
This theorem is referenced by:  dvelimc  2772
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