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Theorem spcimegf 2634
Description: Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimgf.1 𝑥𝐴
spcimgf.2 𝑥𝜓
spcimegf.3 (𝑥 = 𝐴 → (𝜓𝜑))
Assertion
Ref Expression
spcimegf (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))

Proof of Theorem spcimegf
StepHypRef Expression
1 spcimgf.2 . . 3 𝑥𝜓
2 spcimgf.1 . . 3 𝑥𝐴
31, 2spcimegft 2631 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜑)) → (𝐴𝑉 → (𝜓 → ∃𝑥𝜑)))
4 spcimegf.3 . 2 (𝑥 = 𝐴 → (𝜓𝜑))
53, 4mpg 1340 1 (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  wnf 1349  wex 1381  wcel 1393  wnfc 2165
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559
This theorem is referenced by: (None)
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