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Theorem exmidexmid 4115
Description: EXMID implies that an arbitrary proposition is decidable. That is, EXMID captures the usual meaning of excluded middle when stated in terms of propositions.

To get other propositional statements which are equivalent to excluded middle, combine this with notnotrdc 828, peircedc 899, or condc 838.

(Contributed by Jim Kingdon, 18-Jun-2022.)

Assertion
Ref Expression
exmidexmid (EXMIDDECID 𝜑)

Proof of Theorem exmidexmid
Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3177 . . 3 {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}
2 df-exmid 4114 . . . 4 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
3 p0ex 4107 . . . . . 6 {∅} ∈ V
43rabex 4067 . . . . 5 {𝑧 ∈ {∅} ∣ 𝜑} ∈ V
5 sseq1 3115 . . . . . 6 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ⊆ {∅} ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
6 eleq2 2201 . . . . . . 7 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (∅ ∈ 𝑥 ↔ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
76dcbid 823 . . . . . 6 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (DECID ∅ ∈ 𝑥DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
85, 7imbi12d 233 . . . . 5 (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
94, 8spcv 2774 . . . 4 (∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) → ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
102, 9sylbi 120 . . 3 (EXMID → ({𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} → DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
111, 10mpi 15 . 2 (EXMIDDECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})
12 0ex 4050 . . . . 5 ∅ ∈ V
1312snid 3551 . . . 4 ∅ ∈ {∅}
14 biidd 171 . . . . 5 (𝑧 = ∅ → (𝜑𝜑))
1514elrab 2835 . . . 4 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ (∅ ∈ {∅} ∧ 𝜑))
1613, 15mpbiran 924 . . 3 (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
1716dcbii 825 . 2 (DECID ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ DECID 𝜑)
1811, 17sylib 121 1 (EXMIDDECID 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  DECID wdc 819  wal 1329   = wceq 1331  wcel 1480  {crab 2418  wss 3066  c0 3358  {csn 3522  EXMIDwem 4113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093
This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rab 2423  df-v 2683  df-dif 3068  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-exmid 4114
This theorem is referenced by:  exmidn0m  4119  exmid0el  4122  exmidel  4123  exmidundif  4124  exmidundifim  4125  sbthlemi3  6840  sbthlemi5  6842  sbthlemi6  6843  exmidomniim  7006  exmidfodomrlemim  7050
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