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Theorem abrexex2 5988
Description: Existence of an existentially restricted class abstraction. 𝜑 is normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 5981. (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 1491 . . . 4 𝑧𝑥𝐴 𝜑
2 nfcv 2256 . . . . 5 𝑦𝐴
3 nfs1v 1890 . . . . 5 𝑦[𝑧 / 𝑦]𝜑
42, 3nfrexxy 2447 . . . 4 𝑦𝑥𝐴 [𝑧 / 𝑦]𝜑
5 sbequ12 1727 . . . . 5 (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
65rexbidv 2413 . . . 4 (𝑦 = 𝑧 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑))
71, 4, 6cbvab 2238 . . 3 {𝑦 ∣ ∃𝑥𝐴 𝜑} = {𝑧 ∣ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑}
8 df-clab 2102 . . . . 5 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
98rexbii 2417 . . . 4 (∃𝑥𝐴 𝑧 ∈ {𝑦𝜑} ↔ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑)
109abbii 2231 . . 3 {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}} = {𝑧 ∣ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑}
117, 10eqtr4i 2139 . 2 {𝑦 ∣ ∃𝑥𝐴 𝜑} = {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}}
12 df-iun 3783 . . 3 𝑥𝐴 {𝑦𝜑} = {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}}
13 abrexex2.1 . . . 4 𝐴 ∈ V
14 abrexex2.2 . . . 4 {𝑦𝜑} ∈ V
1513, 14iunex 5987 . . 3 𝑥𝐴 {𝑦𝜑} ∈ V
1612, 15eqeltrri 2189 . 2 {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}} ∈ V
1711, 16eqeltri 2188 1 {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V
Colors of variables: wff set class
Syntax hints:  wcel 1463  [wsb 1718  {cab 2101  wrex 2392  Vcvv 2658   ciun 3781
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099
This theorem is referenced by:  abexssex  5989  abexex  5990  oprabrexex2  5994  ab2rexex  5995  ab2rexex2  5996
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