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Theorem riinrab 4041
 Description: Relative intersection of a relative abstraction. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
riinrab (Ax X {y A φ}) = {y A x X φ}
Distinct variable groups:   x,A,y   x,X,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem riinrab
StepHypRef Expression
1 riin0 4039 . . 3 (X = → (Ax X {y A φ}) = A)
2 rzal 3651 . . . . 5 (X = x X φ)
32ralrimivw 2698 . . . 4 (X = y A x X φ)
4 rabid2 2788 . . . 4 (A = {y A x X φ} ↔ y A x X φ)
53, 4sylibr 203 . . 3 (X = A = {y A x X φ})
61, 5eqtrd 2385 . 2 (X = → (Ax X {y A φ}) = {y A x X φ})
7 ssrab2 3351 . . . . 5 {y A φ} A
87rgenw 2681 . . . 4 x X {y A φ} A
9 riinn0 4040 . . . 4 ((x X {y A φ} A X) → (Ax X {y A φ}) = x X {y A φ})
108, 9mpan 651 . . 3 (X → (Ax X {y A φ}) = x X {y A φ})
11 iinrab 4028 . . 3 (Xx X {y A φ} = {y A x X φ})
1210, 11eqtrd 2385 . 2 (X → (Ax X {y A φ}) = {y A x X φ})
136, 12pm2.61ine 2592 1 (Ax X {y A φ}) = {y A x X φ}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ≠ wne 2516  ∀wral 2614  {crab 2618   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550  ∩ciin 3970 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-rex 2620  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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