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Theorem rspcimedv 2673
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 107 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐴)
32eleq1d 2120 . . . . . 6 ((𝜑𝑥 = 𝐴) → (𝑥𝐵𝐴𝐵))
43biimprd 151 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝐴𝐵𝑥𝐵))
5 rspcimedv.2 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
64, 5anim12d 322 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐴𝐵𝜒) → (𝑥𝐵𝜓)))
71, 6spcimedv 2654 . . 3 (𝜑 → ((𝐴𝐵𝜒) → ∃𝑥(𝑥𝐵𝜓)))
81, 7mpand 413 . 2 (𝜑 → (𝜒 → ∃𝑥(𝑥𝐵𝜓)))
9 df-rex 2327 . 2 (∃𝑥𝐵 𝜓 ↔ ∃𝑥(𝑥𝐵𝜓))
108, 9syl6ibr 155 1 (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1257  wex 1395  wcel 1407  wrex 2322
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 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-rex 2327  df-v 2574
This theorem is referenced by:  rspcedv  2675
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