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Theorem r19.3rm 3310
 Description: Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.)
Hypothesis
Ref Expression
r19.3rm.1 𝑥𝜑
Assertion
Ref Expression
r19.3rm (∃𝑦 𝑦𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem r19.3rm
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 biimt 230 . . . 4 (∃𝑥 𝑥𝐴 → (𝜑 ↔ (∃𝑥 𝑥𝐴𝜑)))
6 df-ral 2311 . . . . 5 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
7 r19.3rm.1 . . . . . 6 𝑥𝜑
8719.23 1568 . . . . 5 (∀𝑥(𝑥𝐴𝜑) ↔ (∃𝑥 𝑥𝐴𝜑))
96, 8bitri 173 . . . 4 (∀𝑥𝐴 𝜑 ↔ (∃𝑥 𝑥𝐴𝜑))
105, 9syl6bbr 187 . . 3 (∃𝑥 𝑥𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
114, 10sylbi 114 . 2 (∃𝑎 𝑎𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
122, 11sylbir 125 1 (∃𝑦 𝑦𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  Ⅎwnf 1349  ∃wex 1381   ∈ wcel 1393  ∀wral 2306 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-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-cleq 2033  df-clel 2036  df-ral 2311 This theorem is referenced by:  r19.28m  3311  r19.3rmv  3312  r19.27m  3316  indstr  8536
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