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Theorem acexmidlemv 5453
Description: Lemma for acexmid 5454.

This is acexmid 5454 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 4214 . . . 4 {𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)} On
2 pp0ex 3931 . . . . 5 {∅, {∅}} V
32rabex 3892 . . . 4 {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)} V
4 prexg 3938 . . . 4 (({𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)} On {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)} V) → {{𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)}, {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)}} V)
51, 3, 4mp2an 402 . . 3 {{𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)}, {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)}} V
6 raleq 2499 . . . 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 1703 . . 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 2602 . 2 yz {{𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)}, {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)}}w z ∃!v z u y (z u v u)
10 eqeq1 2043 . . . . . 6 (𝑠 = 𝑡 → (𝑠 = ∅ ↔ 𝑡 = ∅))
1110orbi1d 704 . . . . 5 (𝑠 = 𝑡 → ((𝑠 = ∅ φ) ↔ (𝑡 = ∅ φ)))
1211cbvrabv 2550 . . . 4 {𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)} = {𝑡 {∅, {∅}} ∣ (𝑡 = ∅ φ)}
13 eqeq1 2043 . . . . . 6 (𝑠 = 𝑡 → (𝑠 = {∅} ↔ 𝑡 = {∅}))
1413orbi1d 704 . . . . 5 (𝑠 = 𝑡 → ((𝑠 = {∅} φ) ↔ (𝑡 = {∅} φ)))
1514cbvrabv 2550 . . . 4 {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)} = {𝑡 {∅, {∅}} ∣ (𝑡 = {∅} φ)}
16 eqid 2037 . . . 4 {{𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)}, {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)}} = {{𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)}, {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)}}
1712, 15, 16acexmidlem2 5452 . . 3 (z {{𝑠 {∅, {∅}} ∣ (𝑠 = ∅ φ)}, {𝑠 {∅, {∅}} ∣ (𝑠 = {∅} φ)}}w z ∃!v z u y (z u v u) → (φ ¬ φ))
1817exlimiv 1486 . 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 628   = wceq 1242  wex 1378   wcel 1390  wral 2300  wrex 2301  ∃!wreu 2302  {crab 2304  Vcvv 2551  c0 3218  {csn 3367  {cpr 3368  Oncon0 4066
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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074  df-iota 4810  df-riota 5411
This theorem is referenced by:  acexmid  5454
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