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Theorem dfid3 5158
Description: A stronger version of df-id 5157 that doesn't require 𝑥 and 𝑦 to be distinct. Ordinarily, we wouldn't use this as a definition, since non-distinct dummy variables would make soundness verification more difficult (as the proof here shows). The proof can be instructive in showing how distinct variable requirements may be eliminated, a task that is not necessarily obvious. (Contributed by NM, 5-Feb-2008.) (Revised by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
dfid3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}

Proof of Theorem dfid3
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-id 5157 . 2 I = {⟨𝑥, 𝑧⟩ ∣ 𝑥 = 𝑧}
2 ancom 452 . . . . . . . . . . 11 ((𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ (𝑥 = 𝑧𝑤 = ⟨𝑥, 𝑧⟩))
3 equcom 2103 . . . . . . . . . . . 12 (𝑥 = 𝑧𝑧 = 𝑥)
43anbi1i 602 . . . . . . . . . . 11 ((𝑥 = 𝑧𝑤 = ⟨𝑥, 𝑧⟩) ↔ (𝑧 = 𝑥𝑤 = ⟨𝑥, 𝑧⟩))
52, 4bitri 264 . . . . . . . . . 10 ((𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ (𝑧 = 𝑥𝑤 = ⟨𝑥, 𝑧⟩))
65exbii 1924 . . . . . . . . 9 (∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑧(𝑧 = 𝑥𝑤 = ⟨𝑥, 𝑧⟩))
7 opeq2 4540 . . . . . . . . . . 11 (𝑧 = 𝑥 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑥⟩)
87eqeq2d 2781 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑤 = ⟨𝑥, 𝑧⟩ ↔ 𝑤 = ⟨𝑥, 𝑥⟩))
98equsexvw 2090 . . . . . . . . 9 (∃𝑧(𝑧 = 𝑥𝑤 = ⟨𝑥, 𝑧⟩) ↔ 𝑤 = ⟨𝑥, 𝑥⟩)
10 equid 2097 . . . . . . . . . 10 𝑥 = 𝑥
1110biantru 513 . . . . . . . . 9 (𝑤 = ⟨𝑥, 𝑥⟩ ↔ (𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
126, 9, 113bitri 286 . . . . . . . 8 (∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ (𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
1312exbii 1924 . . . . . . 7 (∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
14 nfe1 2183 . . . . . . . 8 𝑥𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥)
151419.9 2228 . . . . . . 7 (∃𝑥𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ ∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
1613, 15bitr4i 267 . . . . . 6 (∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
17 opeq2 4540 . . . . . . . . . . 11 (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
1817eqeq2d 2781 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑤 = ⟨𝑥, 𝑥⟩ ↔ 𝑤 = ⟨𝑥, 𝑦⟩))
19 equequ2 2111 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 = 𝑥𝑥 = 𝑦))
2018, 19anbi12d 608 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
2120sps 2209 . . . . . . . 8 (∀𝑥 𝑥 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
2221drex1 2477 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
2322drex2 2478 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
2416, 23syl5bb 272 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
25 nfnae 2470 . . . . . 6 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
26 nfnae 2470 . . . . . . 7 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
27 nfcvd 2914 . . . . . . . . 9 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑤)
28 nfcvf2 2938 . . . . . . . . . 10 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)
29 nfcvd 2914 . . . . . . . . . 10 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑧)
3028, 29nfopd 4556 . . . . . . . . 9 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥, 𝑧⟩)
3127, 30nfeqd 2921 . . . . . . . 8 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑤 = ⟨𝑥, 𝑧⟩)
3228, 29nfeqd 2921 . . . . . . . 8 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 = 𝑧)
3331, 32nfand 1978 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧))
34 opeq2 4540 . . . . . . . . . 10 (𝑧 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑦⟩)
3534eqeq2d 2781 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑤 = ⟨𝑥, 𝑧⟩ ↔ 𝑤 = ⟨𝑥, 𝑦⟩))
36 equequ2 2111 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
3735, 36anbi12d 608 . . . . . . . 8 (𝑧 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
3837a1i 11 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦))))
3926, 33, 38cbvexd 2437 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
4025, 39exbid 2247 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
4124, 40pm2.61i 176 . . . 4 (∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦))
4241abbii 2888 . . 3 {𝑤 ∣ ∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧)} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)}
43 df-opab 4847 . . 3 {⟨𝑥, 𝑧⟩ ∣ 𝑥 = 𝑧} = {𝑤 ∣ ∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧)}
44 df-opab 4847 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)}
4542, 43, 443eqtr4i 2803 . 2 {⟨𝑥, 𝑧⟩ ∣ 𝑥 = 𝑧} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
461, 45eqtri 2793 1 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wal 1629   = wceq 1631  wex 1852  {cab 2757  cop 4322  {copab 4846   I cid 5156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-opab 4847  df-id 5157
This theorem is referenced by:  dfid2  5160  reli  5388  opabresid  5596  ider  7932  cnmptid  21685  cossssid2  34560  cossid  34572
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