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Theorem iinrabm 3710
Description: Indexed intersection of a restricted class builder. (Contributed by Jim Kingdon, 16-Aug-2018.)
Assertion
Ref Expression
iinrabm (x x A x A {y Bφ} = {y Bx A 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 3308 . . 3 (x x A → (x A (y B φ) ↔ (y B x A φ)))
21abbidv 2152 . 2 (x x A → {yx A (y B φ)} = {y ∣ (y B x A φ)})
3 df-rab 2309 . . . . 5 {y Bφ} = {y ∣ (y B φ)}
43a1i 9 . . . 4 (x A → {y Bφ} = {y ∣ (y B φ)})
54iineq2i 3667 . . 3 x A {y Bφ} = x A {y ∣ (y B φ)}
6 iinab 3709 . . 3 x A {y ∣ (y B φ)} = {yx A (y B φ)}
75, 6eqtri 2057 . 2 x A {y Bφ} = {yx A (y B φ)}
8 df-rab 2309 . 2 {y Bx A φ} = {y ∣ (y B x A φ)}
92, 7, 83eqtr4g 2094 1 (x x A x A {y Bφ} = {y Bx A φ})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wex 1378   wcel 1390  {cab 2023  wral 2300  {crab 2304   ciin 3649
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-v 2553  df-iin 3651
This theorem is referenced by: (None)
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