Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  abrexex2 GIF version

Theorem abrexex2 5778
 Description: Existence of an existentially restricted class abstraction. 𝜑 is normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 5771. (Contributed by NM, 12-Sep-2004.)
Hypotheses
Ref Expression
abrexex2.1 𝐴 ∈ V
abrexex2.2 {𝑦𝜑} ∈ V
Assertion
Ref Expression
abrexex2 {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem abrexex2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1437 . . . 4 𝑧𝑥𝐴 𝜑
2 nfcv 2194 . . . . 5 𝑦𝐴
3 nfs1v 1831 . . . . 5 𝑦[𝑧 / 𝑦]𝜑
42, 3nfrexxy 2378 . . . 4 𝑦𝑥𝐴 [𝑧 / 𝑦]𝜑
5 sbequ12 1670 . . . . 5 (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
65rexbidv 2344 . . . 4 (𝑦 = 𝑧 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑))
71, 4, 6cbvab 2176 . . 3 {𝑦 ∣ ∃𝑥𝐴 𝜑} = {𝑧 ∣ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑}
8 df-clab 2043 . . . . 5 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
98rexbii 2348 . . . 4 (∃𝑥𝐴 𝑧 ∈ {𝑦𝜑} ↔ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑)
109abbii 2169 . . 3 {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}} = {𝑧 ∣ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑}
117, 10eqtr4i 2079 . 2 {𝑦 ∣ ∃𝑥𝐴 𝜑} = {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}}
12 df-iun 3686 . . 3 𝑥𝐴 {𝑦𝜑} = {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}}
13 abrexex2.1 . . . 4 𝐴 ∈ V
14 abrexex2.2 . . . 4 {𝑦𝜑} ∈ V
1513, 14iunex 5777 . . 3 𝑥𝐴 {𝑦𝜑} ∈ V
1612, 15eqeltrri 2127 . 2 {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}} ∈ V
1711, 16eqeltri 2126 1 {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1409  [wsb 1661  {cab 2042  ∃wrex 2324  Vcvv 2574  ∪ ciun 3684 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937 This theorem is referenced by:  abexssex  5779  abexex  5780  oprabrexex2  5784  ab2rexex  5785  ab2rexex2  5786
 Copyright terms: Public domain W3C validator