Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  axprlem3 Structured version   Visualization version   GIF version

Theorem axprlem3 5291
 Description: Lemma for axpr 5294. Eliminate the antecedent of the relevant replacement instance. (Contributed by Rohan Ridenour, 10-Aug-2023.)
Assertion
Ref Expression
axprlem3 𝑧𝑤(𝑤𝑧 ↔ ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
Distinct variable groups:   𝑥,𝑧,𝑤   𝑦,𝑧,𝑤   𝑧,𝑛,𝑤   𝑧,𝑠,𝑝,𝑤

Proof of Theorem axprlem3
StepHypRef Expression
1 nfv 1915 . . 3 𝑧if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)
21axrep4 5159 . 2 (∀𝑠𝑧𝑤(if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧) → ∃𝑧𝑤(𝑤𝑧 ↔ ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦))))
3 ax6evr 2022 . . . 4 𝑧 𝑥 = 𝑧
4 ifptru 1071 . . . . . . . . 9 (∃𝑛 𝑛𝑠 → (if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) ↔ 𝑤 = 𝑥))
54biimpd 232 . . . . . . . 8 (∃𝑛 𝑛𝑠 → (if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑥))
6 equtrr 2029 . . . . . . . 8 (𝑥 = 𝑧 → (𝑤 = 𝑥𝑤 = 𝑧))
75, 6sylan9r 512 . . . . . . 7 ((𝑥 = 𝑧 ∧ ∃𝑛 𝑛𝑠) → (if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧))
87alrimiv 1928 . . . . . 6 ((𝑥 = 𝑧 ∧ ∃𝑛 𝑛𝑠) → ∀𝑤(if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧))
98expcom 417 . . . . 5 (∃𝑛 𝑛𝑠 → (𝑥 = 𝑧 → ∀𝑤(if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧)))
109eximdv 1918 . . . 4 (∃𝑛 𝑛𝑠 → (∃𝑧 𝑥 = 𝑧 → ∃𝑧𝑤(if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧)))
113, 10mpi 20 . . 3 (∃𝑛 𝑛𝑠 → ∃𝑧𝑤(if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧))
12 ax6evr 2022 . . . 4 𝑧 𝑦 = 𝑧
13 ifpfal 1072 . . . . . . . . . 10 (¬ ∃𝑛 𝑛𝑠 → (if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) ↔ 𝑤 = 𝑦))
1413biimpd 232 . . . . . . . . 9 (¬ ∃𝑛 𝑛𝑠 → (if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑦))
1514adantl 485 . . . . . . . 8 ((𝑦 = 𝑧 ∧ ¬ ∃𝑛 𝑛𝑠) → (if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑦))
16 simpl 486 . . . . . . . 8 ((𝑦 = 𝑧 ∧ ¬ ∃𝑛 𝑛𝑠) → 𝑦 = 𝑧)
17 equtr 2028 . . . . . . . 8 (𝑤 = 𝑦 → (𝑦 = 𝑧𝑤 = 𝑧))
1815, 16, 17syl6ci 71 . . . . . . 7 ((𝑦 = 𝑧 ∧ ¬ ∃𝑛 𝑛𝑠) → (if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧))
1918alrimiv 1928 . . . . . 6 ((𝑦 = 𝑧 ∧ ¬ ∃𝑛 𝑛𝑠) → ∀𝑤(if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧))
2019expcom 417 . . . . 5 (¬ ∃𝑛 𝑛𝑠 → (𝑦 = 𝑧 → ∀𝑤(if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧)))
2120eximdv 1918 . . . 4 (¬ ∃𝑛 𝑛𝑠 → (∃𝑧 𝑦 = 𝑧 → ∃𝑧𝑤(if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧)))
2212, 21mpi 20 . . 3 (¬ ∃𝑛 𝑛𝑠 → ∃𝑧𝑤(if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧))
2311, 22pm2.61i 185 . 2 𝑧𝑤(if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦) → 𝑤 = 𝑧)
242, 23mpg 1799 1 𝑧𝑤(𝑤𝑧 ↔ ∃𝑠(𝑠𝑝 ∧ if-(∃𝑛 𝑛𝑠, 𝑤 = 𝑥, 𝑤 = 𝑦)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  if-wif 1058  ∀wal 1536  ∃wex 1781   ∈ wcel 2111 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-rep 5154 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-tru 1541  df-ex 1782  df-nf 1786 This theorem is referenced by:  axpr  5294
 Copyright terms: Public domain W3C validator