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Theorem onsucelsucexmidlem1 4197
 Description: Lemma for onsucelsucexmid 4199. (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 3858 . . 3 V
21prid1 3450 . 2 {∅, {∅}}
3 eqid 2022 . . 3 ∅ = ∅
43orci 637 . 2 (∅ = ∅ φ)
5 eqeq1 2028 . . . 4 (x = ∅ → (x = ∅ ↔ ∅ = ∅))
65orbi1d 692 . . 3 (x = ∅ → ((x = ∅ φ) ↔ (∅ = ∅ φ)))
76elrab 2675 . 2 (∅ {x {∅, {∅}} ∣ (x = ∅ φ)} ↔ (∅ {∅, {∅}} (∅ = ∅ φ)))
82, 4, 7mpbir2an 837 1 {x {∅, {∅}} ∣ (x = ∅ φ)}
 Colors of variables: wff set class Syntax hints:   ∨ wo 616   = wceq 1228   ∈ wcel 1374  {crab 2288  ∅c0 3201  {csn 3350  {cpr 3351 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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-nul 3857 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-nul 3202  df-sn 3356  df-pr 3357 This theorem is referenced by:  onsucelsucexmidlem  4198  onsucelsucexmid  4199  acexmidlem2  5433
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