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Theorem abexex 5698
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 2309 . . . 4 (x A φx(x A φ))
2 abexex.2 . . . . . 6 (φx A)
32pm4.71ri 372 . . . . 5 (φ ↔ (x A φ))
43exbii 1496 . . . 4 (xφx(x A φ))
51, 4bitr4i 176 . . 3 (x A φxφ)
65abbii 2153 . 2 {yx A φ} = {yxφ}
7 abexex.1 . . 3 A V
8 abexex.3 . . 3 {yφ} V
97, 8abrexex2 5696 . 2 {yx A φ} V
106, 9eqeltrri 2111 1 {yxφ} V
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wex 1381   wcel 1393  {cab 2026  wrex 2304  Vcvv 2554
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3866  ax-sep 3869  ax-pow 3921  ax-pr 3938  ax-un 4139
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-un 2919  df-in 2921  df-ss 2928  df-pw 3356  df-sn 3376  df-pr 3377  df-op 3379  df-uni 3575  df-iun 3653  df-br 3759  df-opab 3813  df-mpt 3814  df-id 4024  df-xp 4297  df-rel 4298  df-cnv 4299  df-co 4300  df-dm 4301  df-rn 4302  df-res 4303  df-ima 4304  df-iota 4813  df-fun 4850  df-fn 4851  df-f 4852  df-f1 4853  df-fo 4854  df-f1o 4855  df-fv 4856
This theorem is referenced by: (None)
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