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Theorem acexmidlemv 5434
Description: Lemma for acexmid 5435.

This is acexmid 5435 with additional distinct variable constraints, most notably between φ and x.

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

Hypothesis
Ref Expression
acexmidlemv.choice yz x w z ∃!v z u y (z u v u)
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 4198 . . . 4 {𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)} On
2 pp0ex 3914 . . . . 5 {∅, {∅}} V
32rabex 3875 . . . 4 {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)} V
4 prexg 3921 . . . 4 (({𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)} On {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)} V) → {{𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)}, {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)}} V)
51, 3, 4mp2an 404 . . 3 {{𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)}, {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)}} V
6 raleq 2483 . . . 4 (x = {{𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)}, {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)}} → (z x w z ∃!v z u y (z u v u) ↔ z {{𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)}, {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)}}w z ∃!v z u y (z u v u)))
76exbidv 1688 . . 3 (x = {{𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)}, {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)}} → (yz x w z ∃!v z u y (z u v u) ↔ yz {{𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)}, {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)}}w z ∃!v z u y (z u v u)))
8 acexmidlemv.choice . . 3 yz x w z ∃!v z u y (z u v u)
95, 7, 8vtocl 2585 . 2 yz {{𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)}, {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)}}w z ∃!v z u y (z u v u)
10 eqeq1 2028 . . . . . 6 (𝑠 = 𝑡 → (𝑠 = ∅ ↔ 𝑡 = ∅))
1110orbi1d 692 . . . . 5 (𝑠 = 𝑡 → ((𝑠 = ∅ φ) ↔ (𝑡 = ∅ φ)))
1211cbvrabv 2534 . . . 4 {𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)} = {𝑡 {∅, {∅}} ∣ (𝑡 = ∅ φ)}
13 eqeq1 2028 . . . . . 6 (𝑠 = 𝑡 → (𝑠 = {∅} ↔ 𝑡 = {∅}))
1413orbi1d 692 . . . . 5 (𝑠 = 𝑡 → ((𝑠 = {∅} φ) ↔ (𝑡 = {∅} φ)))
1514cbvrabv 2534 . . . 4 {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)} = {𝑡 {∅, {∅}} ∣ (𝑡 = {∅} φ)}
16 eqid 2022 . . . 4 {{𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)}, {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)}} = {{𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)}, {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)}}
1712, 15, 16acexmidlem2 5433 . . 3 (z {{𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)}, {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)}}w z ∃!v z u y (z u v u) → (φ ¬ φ))
1817exlimiv 1471 . 2 (yz {{𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)}, {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)}}w z ∃!v z u y (z u v u) → (φ ¬ φ))
199, 18ax-mp 7 1 (φ ¬ φ)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   wo 616   = wceq 1228  wex 1362   wcel 1374  wral 2284  wrex 2285  ∃!wreu 2286  {crab 2288  Vcvv 2535  c0 3201  {csn 3350  {cpr 3351  Oncon0 4049
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-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3or 874  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-tr 3829  df-iord 4052  df-on 4054  df-suc 4057  df-iota 4794  df-riota 5393
This theorem is referenced by:  acexmid  5435
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