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Theorem abrexex2 6072
Description: Existence of an existentially restricted class abstraction. 𝜑 is normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 6065. (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 1508 . . . 4 𝑧𝑥𝐴 𝜑
2 nfcv 2299 . . . . 5 𝑦𝐴
3 nfs1v 1919 . . . . 5 𝑦[𝑧 / 𝑦]𝜑
42, 3nfrexxy 2496 . . . 4 𝑦𝑥𝐴 [𝑧 / 𝑦]𝜑
5 sbequ12 1751 . . . . 5 (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
65rexbidv 2458 . . . 4 (𝑦 = 𝑧 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑))
71, 4, 6cbvab 2281 . . 3 {𝑦 ∣ ∃𝑥𝐴 𝜑} = {𝑧 ∣ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑}
8 df-clab 2144 . . . . 5 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
98rexbii 2464 . . . 4 (∃𝑥𝐴 𝑧 ∈ {𝑦𝜑} ↔ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑)
109abbii 2273 . . 3 {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}} = {𝑧 ∣ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑}
117, 10eqtr4i 2181 . 2 {𝑦 ∣ ∃𝑥𝐴 𝜑} = {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}}
12 df-iun 3851 . . 3 𝑥𝐴 {𝑦𝜑} = {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}}
13 abrexex2.1 . . . 4 𝐴 ∈ V
14 abrexex2.2 . . . 4 {𝑦𝜑} ∈ V
1513, 14iunex 6071 . . 3 𝑥𝐴 {𝑦𝜑} ∈ V
1612, 15eqeltrri 2231 . 2 {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}} ∈ V
1711, 16eqeltri 2230 1 {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V
Colors of variables: wff set class
Syntax hints:  [wsb 1742  wcel 2128  {cab 2143  wrex 2436  Vcvv 2712   ciun 3849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178
This theorem is referenced by:  abexssex  6073  abexex  6074  oprabrexex2  6078  ab2rexex  6079  ab2rexex2  6080
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