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Theorem raaanlem 3326
 Description: Special case of raaan 3327 where 𝐴 is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
Hypotheses
Ref Expression
raaan.1 𝑦𝜑
raaan.2 𝑥𝜓
Assertion
Ref Expression
raaanlem (∃𝑥 𝑥𝐴 → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem raaanlem
StepHypRef Expression
1 eleq1 2100 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
21cbvexv 1795 . . 3 (∃𝑥 𝑥𝐴 ↔ ∃𝑦 𝑦𝐴)
3 raaan.1 . . . . 5 𝑦𝜑
43r19.28m 3311 . . . 4 (∃𝑦 𝑦𝐴 → (∀𝑦𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑦𝐴 𝜓)))
54ralbidv 2326 . . 3 (∃𝑦 𝑦𝐴 → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓)))
62, 5sylbi 114 . 2 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓)))
7 nfcv 2178 . . . 4 𝑥𝐴
8 raaan.2 . . . 4 𝑥𝜓
97, 8nfralxy 2360 . . 3 𝑥𝑦𝐴 𝜓
109r19.27m 3316 . 2 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
116, 10bitrd 177 1 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  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-tru 1246  df-nf 1350  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311 This theorem is referenced by:  raaan  3327
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