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Theorem r19.9rmv 3313
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 2100 . . 3 (𝑎 = 𝑦 → (𝑎𝐴𝑦𝐴))
21cbvexv 1795 . 2 (∃𝑎 𝑎𝐴 ↔ ∃𝑦 𝑦𝐴)
3 eleq1 2100 . . . 4 (𝑎 = 𝑥 → (𝑎𝐴𝑥𝐴))
43cbvexv 1795 . . 3 (∃𝑎 𝑎𝐴 ↔ ∃𝑥 𝑥𝐴)
5 df-rex 2312 . . . . 5 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
6 19.41v 1782 . . . . 5 (∃𝑥(𝑥𝐴𝜑) ↔ (∃𝑥 𝑥𝐴𝜑))
75, 6bitri 173 . . . 4 (∃𝑥𝐴 𝜑 ↔ (∃𝑥 𝑥𝐴𝜑))
87baibr 829 . . 3 (∃𝑥 𝑥𝐴 → (𝜑 ↔ ∃𝑥𝐴 𝜑))
94, 8sylbi 114 . 2 (∃𝑎 𝑎𝐴 → (𝜑 ↔ ∃𝑥𝐴 𝜑))
102, 9sylbir 125 1 (∃𝑦 𝑦𝐴 → (𝜑 ↔ ∃𝑥𝐴 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98  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-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036  df-rex 2312
This theorem is referenced by:  r19.45mv  3315  iunconstm  3665  fconstfvm  5379  ltexprlemloc  6705
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