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Theorem iinrabm 3747
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 3342 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝑦𝐵𝜑) ↔ (𝑦𝐵 ∧ ∀𝑥𝐴 𝜑)))
21abbidv 2171 . 2 (∃𝑥 𝑥𝐴 → {𝑦 ∣ ∀𝑥𝐴 (𝑦𝐵𝜑)} = {𝑦 ∣ (𝑦𝐵 ∧ ∀𝑥𝐴 𝜑)})
3 df-rab 2332 . . . . 5 {𝑦𝐵𝜑} = {𝑦 ∣ (𝑦𝐵𝜑)}
43a1i 9 . . . 4 (𝑥𝐴 → {𝑦𝐵𝜑} = {𝑦 ∣ (𝑦𝐵𝜑)})
54iineq2i 3704 . . 3 𝑥𝐴 {𝑦𝐵𝜑} = 𝑥𝐴 {𝑦 ∣ (𝑦𝐵𝜑)}
6 iinab 3746 . . 3 𝑥𝐴 {𝑦 ∣ (𝑦𝐵𝜑)} = {𝑦 ∣ ∀𝑥𝐴 (𝑦𝐵𝜑)}
75, 6eqtri 2076 . 2 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦 ∣ ∀𝑥𝐴 (𝑦𝐵𝜑)}
8 df-rab 2332 . 2 {𝑦𝐵 ∣ ∀𝑥𝐴 𝜑} = {𝑦 ∣ (𝑦𝐵 ∧ ∀𝑥𝐴 𝜑)}
92, 7, 83eqtr4g 2113 1 (∃𝑥 𝑥𝐴 𝑥𝐴 {𝑦𝐵𝜑} = {𝑦𝐵 ∣ ∀𝑥𝐴 𝜑})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wex 1397  wcel 1409  {cab 2042  wral 2323  {crab 2327   ciin 3686
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-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  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-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rab 2332  df-v 2576  df-iin 3688
This theorem is referenced by: (None)
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