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Theorem r19.9rmv 3354
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 2145 . . 3 (𝑎 = 𝑦 → (𝑎𝐴𝑦𝐴))
21cbvexv 1838 . 2 (∃𝑎 𝑎𝐴 ↔ ∃𝑦 𝑦𝐴)
3 eleq1 2145 . . . 4 (𝑎 = 𝑥 → (𝑎𝐴𝑥𝐴))
43cbvexv 1838 . . 3 (∃𝑎 𝑎𝐴 ↔ ∃𝑥 𝑥𝐴)
5 df-rex 2359 . . . . 5 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
6 19.41v 1825 . . . . 5 (∃𝑥(𝑥𝐴𝜑) ↔ (∃𝑥 𝑥𝐴𝜑))
75, 6bitri 182 . . . 4 (∃𝑥𝐴 𝜑 ↔ (∃𝑥 𝑥𝐴𝜑))
87baibr 863 . . 3 (∃𝑥 𝑥𝐴 → (𝜑 ↔ ∃𝑥𝐴 𝜑))
94, 8sylbi 119 . 2 (∃𝑎 𝑎𝐴 → (𝜑 ↔ ∃𝑥𝐴 𝜑))
102, 9sylbir 133 1 (∃𝑦 𝑦𝐴 → (𝜑 ↔ ∃𝑥𝐴 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wex 1422  wcel 1434  wrex 2354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-cleq 2076  df-clel 2079  df-rex 2359
This theorem is referenced by:  r19.45mv  3356  r19.44mv  3357  iunconstm  3712  fconstfvm  5455  frecabcl  6096  ltexprlemloc  7069  lcmgcdlem  10839
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