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Theorem onsucelsucexmidlem1 4213
Description: Lemma for onsucelsucexmid 4215. (Contributed by Jim Kingdon, 2-Aug-2019.)
Assertion
Ref Expression
onsucelsucexmidlem1 {x {∅, {∅}} ∣ (x = ∅ φ)}
Distinct variable group:   φ,x

Proof of Theorem onsucelsucexmidlem1
StepHypRef Expression
1 0ex 3875 . . 3 V
21prid1 3467 . 2 {∅, {∅}}
3 eqid 2037 . . 3 ∅ = ∅
43orci 649 . 2 (∅ = ∅ φ)
5 eqeq1 2043 . . . 4 (x = ∅ → (x = ∅ ↔ ∅ = ∅))
65orbi1d 704 . . 3 (x = ∅ → ((x = ∅ φ) ↔ (∅ = ∅ φ)))
76elrab 2692 . 2 (∅ {x {∅, {∅}} ∣ (x = ∅ φ)} ↔ (∅ {∅, {∅}} (∅ = ∅ φ)))
82, 4, 7mpbir2an 848 1 {x {∅, {∅}} ∣ (x = ∅ φ)}
Colors of variables: wff set class
Syntax hints:   wo 628   = wceq 1242   wcel 1390  {crab 2304  c0 3218  {csn 3367  {cpr 3368
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-in1 544  ax-in2 545  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-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-nul 3219  df-sn 3373  df-pr 3374
This theorem is referenced by:  onsucelsucexmidlem  4214  onsucelsucexmid  4215  acexmidlem2  5452
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