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Theorem regexmid 4219
 Description: The axiom of foundation implies excluded middle. By foundation (or regularity), we mean the principle that every inhabited set has an element which is minimal (when arranged by ∈). The statement of foundation here is taken from Metamath Proof Explorer's ax-reg, and is identical (modulo one unnecessary quantifier) to the statement of foundation in Theorem "Foundation implies instances of EM" of [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic". For this reason, IZF does not adopt foundation as an axiom and instead replaces it with ax-setind 4220. (Contributed by Jim Kingdon, 3-Sep-2019.)
Hypothesis
Ref Expression
regexmid.1 (y y xy(y x z(z y → ¬ z x)))
Assertion
Ref Expression
regexmid (φ ¬ φ)
Distinct variable group:   φ,x,y,z

Proof of Theorem regexmid
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 eqid 2037 . . 3 {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))}
21regexmidlemm 4217 . 2 y y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))}
3 pp0ex 3931 . . . 4 {∅, {∅}} V
43rabex 3892 . . 3 {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} V
5 eleq2 2098 . . . . 5 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → (y xy {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))}))
65exbidv 1703 . . . 4 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → (y y xy y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))}))
7 eleq2 2098 . . . . . . . . 9 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → (z xz {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))}))
87notbid 591 . . . . . . . 8 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → (¬ z x ↔ ¬ z {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))}))
98imbi2d 219 . . . . . . 7 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → ((z y → ¬ z x) ↔ (z y → ¬ z {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))})))
109albidv 1702 . . . . . 6 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → (z(z y → ¬ z x) ↔ z(z y → ¬ z {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))})))
115, 10anbi12d 442 . . . . 5 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → ((y x z(z y → ¬ z x)) ↔ (y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} z(z y → ¬ z {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))}))))
1211exbidv 1703 . . . 4 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → (y(y x z(z y → ¬ z x)) ↔ y(y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} z(z y → ¬ z {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))}))))
136, 12imbi12d 223 . . 3 (x = {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → ((y y xy(y x z(z y → ¬ z x))) ↔ (y y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → y(y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} z(z y → ¬ z {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))})))))
14 regexmid.1 . . 3 (y y xy(y x z(z y → ¬ z x)))
154, 13, 14vtocl 2602 . 2 (y y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} → y(y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} z(z y → ¬ z {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))})))
161regexmidlem1 4218 . 2 (y(y {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))} z(z y → ¬ z {w {∅, {∅}} ∣ (w = {∅} (w = ∅ φ))})) → (φ ¬ φ))
172, 15, 16mp2b 8 1 (φ ¬ φ)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∨ wo 628  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390  {crab 2304  ∅c0 3218  {csn 3367  {cpr 3368 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-bndl 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 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374 This theorem is referenced by: (None)
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