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Theorem r19.9rmv 3340
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 2116 . . 3 (𝑎 = 𝑦 → (𝑎𝐴𝑦𝐴))
21cbvexv 1811 . 2 (∃𝑎 𝑎𝐴 ↔ ∃𝑦 𝑦𝐴)
3 eleq1 2116 . . . 4 (𝑎 = 𝑥 → (𝑎𝐴𝑥𝐴))
43cbvexv 1811 . . 3 (∃𝑎 𝑎𝐴 ↔ ∃𝑥 𝑥𝐴)
5 df-rex 2329 . . . . 5 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
6 19.41v 1798 . . . . 5 (∃𝑥(𝑥𝐴𝜑) ↔ (∃𝑥 𝑥𝐴𝜑))
75, 6bitri 177 . . . 4 (∃𝑥𝐴 𝜑 ↔ (∃𝑥 𝑥𝐴𝜑))
87baibr 840 . . 3 (∃𝑥 𝑥𝐴 → (𝜑 ↔ ∃𝑥𝐴 𝜑))
94, 8sylbi 118 . 2 (∃𝑎 𝑎𝐴 → (𝜑 ↔ ∃𝑥𝐴 𝜑))
102, 9sylbir 129 1 (∃𝑦 𝑦𝐴 → (𝜑 ↔ ∃𝑥𝐴 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wex 1397  wcel 1409  wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049  df-clel 2052  df-rex 2329
This theorem is referenced by:  r19.45mv  3342  iunconstm  3692  fconstfvm  5406  ltexprlemloc  6762
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