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Theorem spcimegf 2739
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 2736 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜓𝜑)) → (𝐴𝑉 → (𝜓 → ∃𝑥𝜑)))
4 spcimegf.3 . 2 (𝑥 = 𝐴 → (𝜓𝜑))
53, 4mpg 1410 1 (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  wnf 1419  wex 1451  wcel 1463  wnfc 2243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660
This theorem is referenced by: (None)
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