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Theorem iunrab 3695
Description: The indexed union of a restricted class abstraction. (Contributed by NM, 3-Jan-2004.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
iunrab x A {y Bφ} = {y Bx A φ}
Distinct variable groups:   y,A   x,y   x,B
Allowed substitution hints:   φ(x,y)   A(x)   B(y)

Proof of Theorem iunrab
StepHypRef Expression
1 iunab 3694 . 2 x A {y ∣ (y B φ)} = {yx A (y B φ)}
2 df-rab 2309 . . . 4 {y Bφ} = {y ∣ (y B φ)}
32a1i 9 . . 3 (x A → {y Bφ} = {y ∣ (y B φ)})
43iuneq2i 3666 . 2 x A {y Bφ} = x A {y ∣ (y B φ)}
5 df-rab 2309 . . 3 {y Bx A φ} = {y ∣ (y B x A φ)}
6 r19.42v 2461 . . . 4 (x A (y B φ) ↔ (y B x A φ))
76abbii 2150 . . 3 {yx A (y B φ)} = {y ∣ (y B x A φ)}
85, 7eqtr4i 2060 . 2 {y Bx A φ} = {yx A (y B φ)}
91, 4, 83eqtr4i 2067 1 x A {y Bφ} = {y Bx A φ}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242   wcel 1390  {cab 2023  wrex 2301  {crab 2304   ciun 3648
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-rex 2306  df-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-iun 3650
This theorem is referenced by: (None)
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