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Theorem iunrab 3674
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 3673 . 2 x A {y ∣ (y B φ)} = {yx A (y B φ)}
2 df-rab 2289 . . . 4 {y Bφ} = {y ∣ (y B φ)}
32a1i 9 . . 3 (x A → {y Bφ} = {y ∣ (y B φ)})
43iuneq2i 3645 . 2 x A {y Bφ} = x A {y ∣ (y B φ)}
5 df-rab 2289 . . 3 {y Bx A φ} = {y ∣ (y B x A φ)}
6 r19.42v 2441 . . . 4 (x A (y B φ) ↔ (y B x A φ))
76abbii 2131 . . 3 {yx A (y B φ)} = {y ∣ (y B x A φ)}
85, 7eqtr4i 2041 . 2 {y Bx A φ} = {yx A (y B φ)}
91, 4, 83eqtr4i 2048 1 x A {y Bφ} = {y Bx A φ}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1226   wcel 1370  {cab 2004  wrex 2281  {crab 2284   ciun 3627
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-rab 2289  df-v 2533  df-in 2897  df-ss 2904  df-iun 3629
This theorem is referenced by: (None)
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