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Theorem rspcimedv 2704
Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rspcimdv.1 (𝜑𝐴𝐵)
rspcimedv.2 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
Assertion
Ref Expression
rspcimedv (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥   𝜒,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcimedv
StepHypRef Expression
1 rspcimdv.1 . . 3 (𝜑𝐴𝐵)
2 simpr 108 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐴)
32eleq1d 2148 . . . . . 6 ((𝜑𝑥 = 𝐴) → (𝑥𝐵𝐴𝐵))
43biimprd 156 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝐴𝐵𝑥𝐵))
5 rspcimedv.2 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
64, 5anim12d 328 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐴𝐵𝜒) → (𝑥𝐵𝜓)))
71, 6spcimedv 2685 . . 3 (𝜑 → ((𝐴𝐵𝜒) → ∃𝑥(𝑥𝐵𝜓)))
81, 7mpand 420 . 2 (𝜑 → (𝜒 → ∃𝑥(𝑥𝐵𝜓)))
9 df-rex 2355 . 2 (∃𝑥𝐵 𝜓 ↔ ∃𝑥(𝑥𝐵𝜓))
108, 9syl6ibr 160 1 (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wex 1422  wcel 1434  wrex 2350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604
This theorem is referenced by:  rspcedv  2706
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