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Theorem elrabi 2717
Description: Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
Assertion
Ref Expression
elrabi (𝐴 ∈ {𝑥𝑉𝜑} → 𝐴𝑉)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elrabi
StepHypRef Expression
1 clelab 2178 . . 3 (𝐴 ∈ {𝑥 ∣ (𝑥𝑉𝜑)} ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝑉𝜑)))
2 eleq1 2116 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑉𝐴𝑉))
32anbi1d 446 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝑉𝜑) ↔ (𝐴𝑉𝜑)))
43simprbda 369 . . . 4 ((𝑥 = 𝐴 ∧ (𝑥𝑉𝜑)) → 𝐴𝑉)
54exlimiv 1505 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ (𝑥𝑉𝜑)) → 𝐴𝑉)
61, 5sylbi 118 . 2 (𝐴 ∈ {𝑥 ∣ (𝑥𝑉𝜑)} → 𝐴𝑉)
7 df-rab 2332 . 2 {𝑥𝑉𝜑} = {𝑥 ∣ (𝑥𝑉𝜑)}
86, 7eleq2s 2148 1 (𝐴 ∈ {𝑥𝑉𝜑} → 𝐴𝑉)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wex 1397  wcel 1409  {cab 2042  {crab 2327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-rab 2332
This theorem is referenced by:  ordtriexmidlem  4272  ordtri2or2exmidlem  4278  onsucelsucexmidlem  4281  ordsoexmid  4313  reg3exmidlemwe  4330  acexmidlemcase  5534  genpelvl  6667  genpelvu  6668  nnindnn  7024  nnind  8005  ublbneg  8644
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