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Theorem ab2rexex2 6023
 Description: Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥, 𝑦, and 𝑧. Compare abrexex2 6015. (Contributed by NM, 20-Sep-2011.)
Hypotheses
Ref Expression
ab2rexex2.1 𝐴 ∈ V
ab2rexex2.2 𝐵 ∈ V
ab2rexex2.3 {𝑧𝜑} ∈ V
Assertion
Ref Expression
ab2rexex2 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝜑} ∈ V
Distinct variable groups:   𝑥,𝑧,𝐴   𝑦,𝑧,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑦)   𝐵(𝑥)

Proof of Theorem ab2rexex2
StepHypRef Expression
1 ab2rexex2.1 . 2 𝐴 ∈ V
2 ab2rexex2.2 . . 3 𝐵 ∈ V
3 ab2rexex2.3 . . 3 {𝑧𝜑} ∈ V
42, 3abrexex2 6015 . 2 {𝑧 ∣ ∃𝑦𝐵 𝜑} ∈ V
51, 4abrexex2 6015 1 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝜑} ∈ V
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1480  {cab 2123  ∃wrex 2415  Vcvv 2681 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126 This theorem is referenced by: (None)
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