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Theorem abexex 6017
Description: A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)
Hypotheses
Ref Expression
abexex.1 𝐴 ∈ V
abexex.2 (𝜑𝑥𝐴)
abexex.3 {𝑦𝜑} ∈ V
Assertion
Ref Expression
abexex {𝑦 ∣ ∃𝑥𝜑} ∈ V
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem abexex
StepHypRef Expression
1 df-rex 2420 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 abexex.2 . . . . . 6 (𝜑𝑥𝐴)
32pm4.71ri 389 . . . . 5 (𝜑 ↔ (𝑥𝐴𝜑))
43exbii 1584 . . . 4 (∃𝑥𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
51, 4bitr4i 186 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝜑)
65abbii 2253 . 2 {𝑦 ∣ ∃𝑥𝐴 𝜑} = {𝑦 ∣ ∃𝑥𝜑}
7 abexex.1 . . 3 𝐴 ∈ V
8 abexex.3 . . 3 {𝑦𝜑} ∈ V
97, 8abrexex2 6015 . 2 {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V
106, 9eqeltrri 2211 1 {𝑦 ∣ ∃𝑥𝜑} ∈ V
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wex 1468  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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