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Theorem iunrab 4013
 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 B φ} = {y B x 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 4012 . 2 x A {y (y B φ)} = {y x A (y B φ)}
2 df-rab 2623 . . . 4 {y B φ} = {y (y B φ)}
32a1i 10 . . 3 (x A → {y B φ} = {y (y B φ)})
43iuneq2i 3987 . 2 x A {y B φ} = x A {y (y B φ)}
5 df-rab 2623 . . 3 {y B x A φ} = {y (y B x A φ)}
6 r19.42v 2765 . . . 4 (x A (y B φ) ↔ (y B x A φ))
76abbii 2465 . . 3 {y x A (y B φ)} = {y (y B x A φ)}
85, 7eqtr4i 2376 . 2 {y B x A φ} = {y x A (y B φ)}
91, 4, 83eqtr4i 2383 1 x A {y B φ} = {y B x A φ}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  {crab 2618  ∪ciun 3969 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-iun 3971 This theorem is referenced by: (None)
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