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Theorem spcimegf 2830
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 2827 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜑)) → (𝐴𝑉 → (𝜓 → ∃𝑥𝜑)))
4 spcimegf.3 . 2 (𝑥 = 𝐴 → (𝜓𝜑))
53, 4mpg 1461 1 (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  wnf 1470  wex 1502  wcel 2158  wnfc 2316
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-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751
This theorem is referenced by: (None)
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