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Theorem abrexex2 7040
 Description: Existence of an existentially restricted class abstraction. 𝜑 is normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 7033. (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 1830 . . . 4 𝑧𝑥𝐴 𝜑
2 nfcv 2751 . . . . 5 𝑦𝐴
3 nfs1v 2425 . . . . 5 𝑦[𝑧 / 𝑦]𝜑
42, 3nfrex 2990 . . . 4 𝑦𝑥𝐴 [𝑧 / 𝑦]𝜑
5 sbequ12 2097 . . . . 5 (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑))
65rexbidv 3034 . . . 4 (𝑦 = 𝑧 → (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑))
71, 4, 6cbvab 2733 . . 3 {𝑦 ∣ ∃𝑥𝐴 𝜑} = {𝑧 ∣ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑}
8 df-clab 2597 . . . . 5 (𝑧 ∈ {𝑦𝜑} ↔ [𝑧 / 𝑦]𝜑)
98rexbii 3023 . . . 4 (∃𝑥𝐴 𝑧 ∈ {𝑦𝜑} ↔ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑)
109abbii 2726 . . 3 {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}} = {𝑧 ∣ ∃𝑥𝐴 [𝑧 / 𝑦]𝜑}
117, 10eqtr4i 2635 . 2 {𝑦 ∣ ∃𝑥𝐴 𝜑} = {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}}
12 df-iun 4457 . . 3 𝑥𝐴 {𝑦𝜑} = {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}}
13 abrexex2.1 . . . 4 𝐴 ∈ V
14 abrexex2.2 . . . 4 {𝑦𝜑} ∈ V
1513, 14iunex 7039 . . 3 𝑥𝐴 {𝑦𝜑} ∈ V
1612, 15eqeltrri 2685 . 2 {𝑧 ∣ ∃𝑥𝐴 𝑧 ∈ {𝑦𝜑}} ∈ V
1711, 16eqeltri 2684 1 {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V
 Colors of variables: wff setvar class Syntax hints:  [wsb 1867   ∈ wcel 1977  {cab 2596  ∃wrex 2897  Vcvv 3173  ∪ ciun 4455 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812 This theorem is referenced by:  abexssex  7041  abexex  7042  oprabrexex2  7049  ab2rexex  7050  ab2rexex2  7051
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