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Theorem abexex 5695
 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 (φx A)
abexex.3 {yφ} V
Assertion
Ref Expression
abexex {yxφ} V
Distinct variable group:   x,y,A
Allowed substitution hints:   φ(x,y)

Proof of Theorem abexex
StepHypRef Expression
1 df-rex 2306 . . . 4 (x A φx(x A φ))
2 abexex.2 . . . . . 6 (φx A)
32pm4.71ri 372 . . . . 5 (φ ↔ (x A φ))
43exbii 1493 . . . 4 (xφx(x A φ))
51, 4bitr4i 176 . . 3 (x A φxφ)
65abbii 2150 . 2 {yx A φ} = {yxφ}
7 abexex.1 . . 3 A V
8 abexex.3 . . 3 {yφ} V
97, 8abrexex2 5693 . 2 {yx A φ} V
106, 9eqeltrri 2108 1 {yxφ} V
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1378   ∈ wcel 1390  {cab 2023  ∃wrex 2301  Vcvv 2551 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853 This theorem is referenced by: (None)
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