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Theorem raaanlem 3354
Description: Special case of raaan 3355 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 2116 . . . 4 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
21cbvexv 1811 . . 3 (∃𝑥 𝑥𝐴 ↔ ∃𝑦 𝑦𝐴)
3 raaan.1 . . . . 5 𝑦𝜑
43r19.28m 3339 . . . 4 (∃𝑦 𝑦𝐴 → (∀𝑦𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑦𝐴 𝜓)))
54ralbidv 2343 . . 3 (∃𝑦 𝑦𝐴 → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓)))
62, 5sylbi 118 . 2 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓)))
7 nfcv 2194 . . . 4 𝑥𝐴
8 raaan.2 . . . 4 𝑥𝜓
97, 8nfralxy 2377 . . 3 𝑥𝑦𝐴 𝜓
109r19.27m 3344 . 2 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝜑 ∧ ∀𝑦𝐴 𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
116, 10bitrd 181 1 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wnf 1365  wex 1397  wcel 1409  wral 2323
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-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328
This theorem is referenced by:  raaan  3355
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