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Theorem elrabi 2689
Description: Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
Assertion
Ref Expression
elrabi (A {x 𝑉φ} → A 𝑉)
Distinct variable groups:   x,A   x,𝑉
Allowed substitution hint:   φ(x)

Proof of Theorem elrabi
StepHypRef Expression
1 clelab 2159 . . 3 (A {x ∣ (x 𝑉 φ)} ↔ x(x = A (x 𝑉 φ)))
2 eleq1 2097 . . . . . 6 (x = A → (x 𝑉A 𝑉))
32anbi1d 438 . . . . 5 (x = A → ((x 𝑉 φ) ↔ (A 𝑉 φ)))
43simprbda 365 . . . 4 ((x = A (x 𝑉 φ)) → A 𝑉)
54exlimiv 1486 . . 3 (x(x = A (x 𝑉 φ)) → A 𝑉)
61, 5sylbi 114 . 2 (A {x ∣ (x 𝑉 φ)} → A 𝑉)
7 df-rab 2309 . 2 {x 𝑉φ} = {x ∣ (x 𝑉 φ)}
86, 7eleq2s 2129 1 (A {x 𝑉φ} → A 𝑉)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wex 1378   wcel 1390  {cab 2023  {crab 2304
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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-rab 2309
This theorem is referenced by:  ordtriexmidlem  4208  onsucelsucexmidlem  4214  ordsoexmid  4240  acexmidlemcase  5450  genpelvl  6494  genpelvu  6495  nnind  7671  ublbneg  8284
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