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Theorem abrexex2g 5689
 Description: Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
abrexex2g ((A 𝑉 x A {yφ} 𝑊) → {yx A φ} V)
Distinct variable groups:   x,A,y   x,𝑉,y   x,𝑊,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem abrexex2g
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 nfv 1418 . . . 4 zx A φ
2 nfcv 2175 . . . . 5 yA
3 nfs1v 1812 . . . . 5 y[z / y]φ
42, 3nfrexxy 2355 . . . 4 yx A [z / y]φ
5 sbequ12 1651 . . . . 5 (y = z → (φ ↔ [z / y]φ))
65rexbidv 2321 . . . 4 (y = z → (x A φx A [z / y]φ))
71, 4, 6cbvab 2157 . . 3 {yx A φ} = {zx A [z / y]φ}
8 df-clab 2024 . . . . 5 (z {yφ} ↔ [z / y]φ)
98rexbii 2325 . . . 4 (x A z {yφ} ↔ x A [z / y]φ)
109abbii 2150 . . 3 {zx A z {yφ}} = {zx A [z / y]φ}
117, 10eqtr4i 2060 . 2 {yx A φ} = {zx A z {yφ}}
12 df-iun 3650 . . 3 x A {yφ} = {zx A z {yφ}}
13 iunexg 5688 . . 3 ((A 𝑉 x A {yφ} 𝑊) → x A {yφ} V)
1412, 13syl5eqelr 2122 . 2 ((A 𝑉 x A {yφ} 𝑊) → {zx A z {yφ}} V)
1511, 14syl5eqel 2121 1 ((A 𝑉 x A {yφ} 𝑊) → {yx A φ} V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  [wsb 1642  {cab 2023  ∀wral 2300  ∃wrex 2301  Vcvv 2551  ∪ ciun 3648 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853 This theorem is referenced by:  frecabex  5923
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