ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rspcimedv GIF version

Theorem rspcimedv 2738
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 2163 . . . . . 6 ((𝜑𝑥 = 𝐴) → (𝑥𝐵𝐴𝐵))
43biimprd 157 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝐴𝐵𝑥𝐵))
5 rspcimedv.2 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
64, 5anim12d 329 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐴𝐵𝜒) → (𝑥𝐵𝜓)))
71, 6spcimedv 2719 . . 3 (𝜑 → ((𝐴𝐵𝜒) → ∃𝑥(𝑥𝐵𝜓)))
81, 7mpand 421 . 2 (𝜑 → (𝜒 → ∃𝑥(𝑥𝐵𝜓)))
9 df-rex 2376 . 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 1296  wex 1433  wcel 1445  wrex 2371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rex 2376  df-v 2635
This theorem is referenced by:  rspcedv  2740
  Copyright terms: Public domain W3C validator