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Theorem spcimedv 2744
Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimdv.1 (𝜑𝐴𝐵)
spcimedv.2 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
Assertion
Ref Expression
spcimedv (𝜑 → (𝜒 → ∃𝑥𝜓))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)

Proof of Theorem spcimedv
StepHypRef Expression
1 spcimedv.2 . . . 4 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
21ex 114 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜒𝜓)))
32alrimiv 1828 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜒𝜓)))
4 spcimdv.1 . 2 (𝜑𝐴𝐵)
5 nfv 1491 . . 3 𝑥𝜒
6 nfcv 2256 . . 3 𝑥𝐴
75, 6spcimegft 2736 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜒𝜓)) → (𝐴𝐵 → (𝜒 → ∃𝑥𝜓)))
83, 4, 7sylc 62 1 (𝜑 → (𝜒 → ∃𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1312   = wceq 1314  wex 1451  wcel 1463
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:  rspcimedv  2763  fihashf1rn  10475
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