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Theorem iinrabm 3719
 Description: Indexed intersection of a restricted class builder. (Contributed by Jim Kingdon, 16-Aug-2018.)
Assertion
Ref Expression
iinrabm (∃𝑥 𝑥𝐴 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∀𝑥𝐴 𝜑})
Distinct variable groups:   𝑦,𝐴,𝑥   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑦)

Proof of Theorem iinrabm
StepHypRef Expression
1 r19.28mv 3314 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝑦𝐵𝜑) ↔ (𝑦𝐵 ∧ ∀𝑥𝐴 𝜑)))
21abbidv 2155 . 2 (∃𝑥 𝑥𝐴 → {𝑦 ∣ ∀𝑥𝐴 (𝑦𝐵𝜑)} = {𝑦 ∣ (𝑦𝐵 ∧ ∀𝑥𝐴 𝜑)})
3 df-rab 2315 . . . . 5 {𝑦𝐵𝜑} = {𝑦 ∣ (𝑦𝐵𝜑)}
43a1i 9 . . . 4 (𝑥𝐴 → {𝑦𝐵𝜑} = {𝑦 ∣ (𝑦𝐵𝜑)})
54iineq2i 3676 . . 3 𝑥𝐴 {𝑦𝐵𝜑} = 𝑥𝐴 {𝑦 ∣ (𝑦𝐵𝜑)}
6 iinab 3718 . . 3 𝑥𝐴 {𝑦 ∣ (𝑦𝐵𝜑)} = {𝑦 ∣ ∀𝑥𝐴 (𝑦𝐵𝜑)}
75, 6eqtri 2060 . 2 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦 ∣ ∀𝑥𝐴 (𝑦𝐵𝜑)}
8 df-rab 2315 . 2 {𝑦𝐵 ∣ ∀𝑥𝐴 𝜑} = {𝑦 ∣ (𝑦𝐵 ∧ ∀𝑥𝐴 𝜑)}
92, 7, 83eqtr4g 2097 1 (∃𝑥 𝑥𝐴 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∀𝑥𝐴 𝜑})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  {cab 2026  ∀wral 2306  {crab 2310  ∩ ciin 3658 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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  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-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-iin 3660 This theorem is referenced by: (None)
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