ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  r19.9rmv GIF version

Theorem r19.9rmv 3454
Description: Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.)
Assertion
Ref Expression
r19.9rmv (∃𝑦 𝑦𝐴 → (𝜑 ↔ ∃𝑥𝐴 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝜑,𝑥
Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem r19.9rmv
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2202 . . 3 (𝑎 = 𝑦 → (𝑎𝐴𝑦𝐴))
21cbvexv 1890 . 2 (∃𝑎 𝑎𝐴 ↔ ∃𝑦 𝑦𝐴)
3 eleq1 2202 . . . 4 (𝑎 = 𝑥 → (𝑎𝐴𝑥𝐴))
43cbvexv 1890 . . 3 (∃𝑎 𝑎𝐴 ↔ ∃𝑥 𝑥𝐴)
5 df-rex 2422 . . . . 5 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
6 19.41v 1874 . . . . 5 (∃𝑥(𝑥𝐴𝜑) ↔ (∃𝑥 𝑥𝐴𝜑))
75, 6bitri 183 . . . 4 (∃𝑥𝐴 𝜑 ↔ (∃𝑥 𝑥𝐴𝜑))
87baibr 905 . . 3 (∃𝑥 𝑥𝐴 → (𝜑 ↔ ∃𝑥𝐴 𝜑))
94, 8sylbi 120 . 2 (∃𝑎 𝑎𝐴 → (𝜑 ↔ ∃𝑥𝐴 𝜑))
102, 9sylbir 134 1 (∃𝑦 𝑦𝐴 → (𝜑 ↔ ∃𝑥𝐴 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wex 1468  wcel 1480  wrex 2417
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-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-clel 2135  df-rex 2422
This theorem is referenced by:  r19.45mv  3456  r19.44mv  3457  iunconstm  3821  fconstfvm  5638  frecabcl  6296  ltexprlemloc  7427  lcmgcdlem  11769
  Copyright terms: Public domain W3C validator