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Theorem acexmidlemb 5424
Description: Lemma for acexmid 5431. (Contributed by Jim Kingdon, 6-Aug-2019.)
Hypotheses
Ref Expression
acexmidlem.a A = {x {∅, {∅}} ∣ (x = ∅ φ)}
acexmidlem.b B = {x {∅, {∅}} ∣ (x = {∅} φ)}
acexmidlem.c 𝐶 = {A, B}
Assertion
Ref Expression
acexmidlemb (∅ Bφ)
Distinct variable groups:   x,A   x,B   x,𝐶   φ,x

Proof of Theorem acexmidlemb
StepHypRef Expression
1 acexmidlem.b . . . 4 B = {x {∅, {∅}} ∣ (x = {∅} φ)}
21eleq2i 2082 . . 3 (∅ B ↔ ∅ {x {∅, {∅}} ∣ (x = {∅} φ)})
3 0ex 3854 . . . . 5 V
43prid1 3446 . . . 4 {∅, {∅}}
5 eqeq1 2024 . . . . . 6 (x = ∅ → (x = {∅} ↔ ∅ = {∅}))
65orbi1d 692 . . . . 5 (x = ∅ → ((x = {∅} φ) ↔ (∅ = {∅} φ)))
76elrab3 2672 . . . 4 (∅ {∅, {∅}} → (∅ {x {∅, {∅}} ∣ (x = {∅} φ)} ↔ (∅ = {∅} φ)))
84, 7ax-mp 7 . . 3 (∅ {x {∅, {∅}} ∣ (x = {∅} φ)} ↔ (∅ = {∅} φ))
92, 8bitri 173 . 2 (∅ B ↔ (∅ = {∅} φ))
10 noel 3201 . . . 4 ¬ ∅
113snid 3373 . . . . 5 {∅}
12 eleq2 2079 . . . . 5 (∅ = {∅} → (∅ ∅ ↔ ∅ {∅}))
1311, 12mpbiri 157 . . . 4 (∅ = {∅} → ∅ ∅)
1410, 13mto 575 . . 3 ¬ ∅ = {∅}
15 orel1 631 . . 3 (¬ ∅ = {∅} → ((∅ = {∅} φ) → φ))
1614, 15ax-mp 7 . 2 ((∅ = {∅} φ) → φ)
179, 16sylbi 114 1 (∅ Bφ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98   wo 616   = wceq 1226   wcel 1370  {crab 2284  c0 3197  {csn 3346  {cpr 3347
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-nul 3853
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-rab 2289  df-v 2533  df-dif 2893  df-un 2895  df-nul 3198  df-sn 3352  df-pr 3353
This theorem is referenced by:  acexmidlem1  5428
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