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Theorem spcimedv 2639
 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 108 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜒𝜓)))
32alrimiv 1754 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜒𝜓)))
4 spcimdv.1 . 2 (𝜑𝐴𝐵)
5 nfv 1421 . . 3 𝑥𝜒
6 nfcv 2178 . . 3 𝑥𝐴
75, 6spcimegft 2631 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜒𝜓)) → (𝐴𝐵 → (𝜒 → ∃𝑥𝜓)))
83, 4, 7sylc 56 1 (𝜑 → (𝜒 → ∃𝑥𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241   = wceq 1243  ∃wex 1381   ∈ wcel 1393 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:  rspcimedv  2658
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