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Theorem rspcimedv 2797
 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 109 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐴)
32eleq1d 2210 . . . . . 6 ((𝜑𝑥 = 𝐴) → (𝑥𝐵𝐴𝐵))
43biimprd 157 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝐴𝐵𝑥𝐵))
5 rspcimedv.2 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
64, 5anim12d 333 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐴𝐵𝜒) → (𝑥𝐵𝜓)))
71, 6spcimedv 2777 . . 3 (𝜑 → ((𝐴𝐵𝜒) → ∃𝑥(𝑥𝐵𝜓)))
81, 7mpand 426 . 2 (𝜑 → (𝜒 → ∃𝑥(𝑥𝐵𝜓)))
9 df-rex 2424 . 2 (∃𝑥𝐵 𝜓 ↔ ∃𝑥(𝑥𝐵𝜓))
108, 9syl6ibr 161 1 (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332  ∃wex 1469   ∈ wcel 2112  ∃wrex 2419 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2123 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1732  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-rex 2424  df-v 2693 This theorem is referenced by:  rspcedv  2799
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