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Theorem oprabidlem 5680
Description: Slight elaboration of exdistrfor 1728. A lemma for oprabid 5681. (Contributed by Jim Kingdon, 15-Jan-2019.)
Assertion
Ref Expression
oprabidlem (∃𝑥𝑦(𝑥 = 𝑧𝜓) → ∃𝑥(𝑥 = 𝑧 ∧ ∃𝑦𝜓))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem oprabidlem
StepHypRef Expression
1 ax-bndl 1444 . . 3 (∀𝑦 𝑦 = 𝑥 ∨ (∀𝑦 𝑦 = 𝑧 ∨ ∀𝑥𝑦(𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)))
2 ax-10 1441 . . . 4 (∀𝑦 𝑦 = 𝑥 → ∀𝑥 𝑥 = 𝑦)
3 dtru 4376 . . . . . 6 ¬ ∀𝑦 𝑦 = 𝑧
4 pm2.53 676 . . . . . 6 ((∀𝑦 𝑦 = 𝑧 ∨ ∀𝑥𝑦(𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)) → (¬ ∀𝑦 𝑦 = 𝑧 → ∀𝑥𝑦(𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)))
53, 4mpi 15 . . . . 5 ((∀𝑦 𝑦 = 𝑧 ∨ ∀𝑥𝑦(𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)) → ∀𝑥𝑦(𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
6 df-nf 1395 . . . . . 6 (Ⅎ𝑦 𝑥 = 𝑧 ↔ ∀𝑦(𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
76albii 1404 . . . . 5 (∀𝑥𝑦 𝑥 = 𝑧 ↔ ∀𝑥𝑦(𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))
85, 7sylibr 132 . . . 4 ((∀𝑦 𝑦 = 𝑧 ∨ ∀𝑥𝑦(𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)) → ∀𝑥𝑦 𝑥 = 𝑧)
92, 8orim12i 711 . . 3 ((∀𝑦 𝑦 = 𝑥 ∨ (∀𝑦 𝑦 = 𝑧 ∨ ∀𝑥𝑦(𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))) → (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥𝑦 𝑥 = 𝑧))
101, 9ax-mp 7 . 2 (∀𝑥 𝑥 = 𝑦 ∨ ∀𝑥𝑦 𝑥 = 𝑧)
1110exdistrfor 1728 1 (∃𝑥𝑦(𝑥 = 𝑧𝜓) → ∃𝑥(𝑥 = 𝑧 ∧ ∃𝑦𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wo 664  wal 1287  wnf 1394  wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-setind 4353
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452
This theorem is referenced by:  oprabid  5681
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