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Theorem iinrab 4028
 Description: Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)
Assertion
Ref Expression
iinrab (Ax A {y B φ} = {y B x A φ})
Distinct variable groups:   y,A,x   x,B
Allowed substitution hints:   φ(x,y)   B(y)

Proof of Theorem iinrab
StepHypRef Expression
1 r19.28zv 3645 . . 3 (A → (x A (y B φ) ↔ (y B x A φ)))
21abbidv 2467 . 2 (A → {y x A (y B φ)} = {y (y B x A φ)})
3 df-rab 2623 . . . . 5 {y B φ} = {y (y B φ)}
43a1i 10 . . . 4 (x A → {y B φ} = {y (y B φ)})
54iineq2i 3988 . . 3 x A {y B φ} = x A {y (y B φ)}
6 iinab 4027 . . 3 x A {y (y B φ)} = {y x A (y B φ)}
75, 6eqtri 2373 . 2 x A {y B φ} = {y x A (y B φ)}
8 df-rab 2623 . 2 {y B x A φ} = {y (y B x A φ)}
92, 7, 83eqtr4g 2410 1 (Ax A {y B φ} = {y B x A φ})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339   ≠ wne 2516  ∀wral 2614  {crab 2618  ∅c0 3550  ∩ciin 3970 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-ne 2518  df-ral 2619  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551  df-iin 3972 This theorem is referenced by:  iinrab2  4029  riinrab  4041
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