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Theorem spcimedv 2656
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 112 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜒𝜓)))
32alrimiv 1770 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜒𝜓)))
4 spcimdv.1 . 2 (𝜑𝐴𝐵)
5 nfv 1437 . . 3 𝑥𝜒
6 nfcv 2194 . . 3 𝑥𝐴
75, 6spcimegft 2648 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜒𝜓)) → (𝐴𝐵 → (𝜒 → ∃𝑥𝜓)))
83, 4, 7sylc 60 1 (𝜑 → (𝜒 → ∃𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257   = wceq 1259  wex 1397  wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  rspcimedv  2675
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