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Theorem spcegf 2690
 Description: Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)
Hypotheses
Ref Expression
spcgf.1 𝑥𝐴
spcgf.2 𝑥𝜓
spcgf.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
spcegf (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))

Proof of Theorem spcegf
StepHypRef Expression
1 spcgf.2 . . 3 𝑥𝜓
2 spcgf.1 . . 3 𝑥𝐴
31, 2spcegft 2686 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝑉 → (𝜓 → ∃𝑥𝜑)))
4 spcgf.3 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
53, 4mpg 1381 1 (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 103   = wceq 1285  Ⅎwnf 1390  ∃wex 1422   ∈ wcel 1434  Ⅎwnfc 2210 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612 This theorem is referenced by:  spcegv  2695  rspce  2705  euotd  4037
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