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Theorem iinrab2 4029
 Description: Indexed intersection of a restricted class builder. (Contributed by NM, 6-Dec-2011.)
Assertion
Ref Expression
iinrab2 (x A {y B φ} ∩ B) = {y B x A φ}
Distinct variable groups:   y,A,x   x,B,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem iinrab2
StepHypRef Expression
1 iineq1 3983 . . . . . 6 (A = x A {y B φ} = x {y B φ})
2 0iin 4024 . . . . . 6 x {y B φ} = V
31, 2syl6eq 2401 . . . . 5 (A = x A {y B φ} = V)
43ineq1d 3456 . . . 4 (A = → (x A {y B φ} ∩ B) = (V ∩ B))
5 incom 3448 . . . . 5 (V ∩ B) = (B ∩ V)
6 inv1 3577 . . . . 5 (B ∩ V) = B
75, 6eqtri 2373 . . . 4 (V ∩ B) = B
84, 7syl6eq 2401 . . 3 (A = → (x A {y B φ} ∩ B) = B)
9 rzal 3651 . . . 4 (A = x A y B φ)
10 rabid2 2788 . . . . 5 (B = {y B x A φ} ↔ y B x A φ)
11 ralcom 2771 . . . . 5 (y B x A φx A y B φ)
1210, 11bitr2i 241 . . . 4 (x A y B φB = {y B x A φ})
139, 12sylib 188 . . 3 (A = B = {y B x A φ})
148, 13eqtrd 2385 . 2 (A = → (x A {y B φ} ∩ B) = {y B x A φ})
15 iinrab 4028 . . . 4 (Ax A {y B φ} = {y B x A φ})
1615ineq1d 3456 . . 3 (A → (x A {y B φ} ∩ B) = ({y B x A φ} ∩ B))
17 ssrab2 3351 . . . 4 {y B x A φ} B
18 dfss 3260 . . . 4 ({y B x A φ} B ↔ {y B x A φ} = ({y B x A φ} ∩ B))
1917, 18mpbi 199 . . 3 {y B x A φ} = ({y B x A φ} ∩ B)
2016, 19syl6eqr 2403 . 2 (A → (x A {y B φ} ∩ B) = {y B x A φ})
2114, 20pm2.61ine 2592 1 (x A {y B φ} ∩ B) = {y B x A φ}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ≠ wne 2516  ∀wral 2614  {crab 2618  Vcvv 2859   ∩ cin 3208   ⊆ wss 3257  ∅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-ss 3259  df-nul 3551  df-iin 3972 This theorem is referenced by: (None)
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