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Theorem acexmidlemv 5873
Description: Lemma for acexmid 5874.

This is acexmid 5874 with additional disjoint variable conditions, most notably between 𝜑 and 𝑥.

(Contributed by Jim Kingdon, 6-Aug-2019.)

Hypothesis
Ref Expression
acexmidlemv.choice 𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)
Assertion
Ref Expression
acexmidlemv (𝜑 ∨ ¬ 𝜑)
Distinct variable group:   𝜑,𝑥,𝑦,𝑧,𝑤,𝑣,𝑢

Proof of Theorem acexmidlemv
Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 onsucelsucexmidlem 4529 . . . 4 {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)} ∈ On
2 pp0ex 4190 . . . . 5 {∅, {∅}} ∈ V
32rabex 4148 . . . 4 {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)} ∈ V
4 prexg 4212 . . . 4 (({𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)} ∈ On ∧ {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)} ∈ V) → {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}} ∈ V)
51, 3, 4mp2an 426 . . 3 {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}} ∈ V
6 raleq 2673 . . . 4 (𝑥 = {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}} → (∀𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∀𝑧 ∈ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}}∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
76exbidv 1825 . . 3 (𝑥 = {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}} → (∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑦𝑧 ∈ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}}∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)))
8 acexmidlemv.choice . . 3 𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)
95, 7, 8vtocl 2792 . 2 𝑦𝑧 ∈ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}}∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢)
10 eqeq1 2184 . . . . . 6 (𝑠 = 𝑡 → (𝑠 = ∅ ↔ 𝑡 = ∅))
1110orbi1d 791 . . . . 5 (𝑠 = 𝑡 → ((𝑠 = ∅ ∨ 𝜑) ↔ (𝑡 = ∅ ∨ 𝜑)))
1211cbvrabv 2737 . . . 4 {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)} = {𝑡 ∈ {∅, {∅}} ∣ (𝑡 = ∅ ∨ 𝜑)}
13 eqeq1 2184 . . . . . 6 (𝑠 = 𝑡 → (𝑠 = {∅} ↔ 𝑡 = {∅}))
1413orbi1d 791 . . . . 5 (𝑠 = 𝑡 → ((𝑠 = {∅} ∨ 𝜑) ↔ (𝑡 = {∅} ∨ 𝜑)))
1514cbvrabv 2737 . . . 4 {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)} = {𝑡 ∈ {∅, {∅}} ∣ (𝑡 = {∅} ∨ 𝜑)}
16 eqid 2177 . . . 4 {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}} = {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}}
1712, 15, 16acexmidlem2 5872 . . 3 (∀𝑧 ∈ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}}∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → (𝜑 ∨ ¬ 𝜑))
1817exlimiv 1598 . 2 (∃𝑦𝑧 ∈ {{𝑠 ∈ {∅, {∅}} ∣ (𝑠 = ∅ ∨ 𝜑)}, {𝑠 ∈ {∅, {∅}} ∣ (𝑠 = {∅} ∨ 𝜑)}}∀𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) → (𝜑 ∨ ¬ 𝜑))
199, 18ax-mp 5 1 (𝜑 ∨ ¬ 𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104  wo 708   = wceq 1353  wex 1492  wcel 2148  wral 2455  wrex 2456  ∃!wreu 2457  {crab 2459  Vcvv 2738  c0 3423  {csn 3593  {cpr 3594  Oncon0 4364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-uni 3811  df-tr 4103  df-iord 4367  df-on 4369  df-suc 4372  df-iota 5179  df-riota 5831
This theorem is referenced by:  acexmid  5874
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