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Theorem r19.9rmv 3393
 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 2157 . . 3 (𝑎 = 𝑦 → (𝑎𝐴𝑦𝐴))
21cbvexv 1850 . 2 (∃𝑎 𝑎𝐴 ↔ ∃𝑦 𝑦𝐴)
3 eleq1 2157 . . . 4 (𝑎 = 𝑥 → (𝑎𝐴𝑥𝐴))
43cbvexv 1850 . . 3 (∃𝑎 𝑎𝐴 ↔ ∃𝑥 𝑥𝐴)
5 df-rex 2376 . . . . 5 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
6 19.41v 1837 . . . . 5 (∃𝑥(𝑥𝐴𝜑) ↔ (∃𝑥 𝑥𝐴𝜑))
75, 6bitri 183 . . . 4 (∃𝑥𝐴 𝜑 ↔ (∃𝑥 𝑥𝐴𝜑))
87baibr 870 . . 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 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-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-ext 2077 This theorem depends on definitions:  df-bi 116  df-cleq 2088  df-clel 2091  df-rex 2376 This theorem is referenced by:  r19.45mv  3395  r19.44mv  3396  iunconstm  3760  fconstfvm  5554  frecabcl  6202  ltexprlemloc  7263  lcmgcdlem  11486
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