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Theorem rspcimedv 2658
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 103 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐴)
32eleq1d 2106 . . . . . 6 ((𝜑𝑥 = 𝐴) → (𝑥𝐵𝐴𝐵))
43biimprd 147 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝐴𝐵𝑥𝐵))
5 rspcimedv.2 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
64, 5anim12d 318 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐴𝐵𝜒) → (𝑥𝐵𝜓)))
71, 6spcimedv 2639 . . 3 (𝜑 → ((𝐴𝐵𝜒) → ∃𝑥(𝑥𝐵𝜓)))
81, 7mpand 405 . 2 (𝜑 → (𝜒 → ∃𝑥(𝑥𝐵𝜓)))
9 df-rex 2312 . 2 (∃𝑥𝐵 𝜓 ↔ ∃𝑥(𝑥𝐵𝜓))
108, 9syl6ibr 151 1 (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97   = wceq 1243  wex 1381  wcel 1393  wrex 2307
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-rex 2312  df-v 2559
This theorem is referenced by:  rspcedv  2660
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