Users' Mathboxes Mathbox for Steven Nguyen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  euabsn2w Structured version   Visualization version   GIF version

Theorem euabsn2w 42689
Description: Replace ax-10 2141, ax-11 2157, ax-12 2177 in euabsn2 4725 with substitution hypotheses. (Contributed by SN, 27-May-2025.)
Hypotheses
Ref Expression
absnw.y (𝑥 = 𝑦 → (𝜑𝜓))
euabsn2w.z (𝑥 = 𝑧 → (𝜑𝜃))
Assertion
Ref Expression
euabsn2w (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦   𝜃,𝑥   𝜑,𝑧   𝑥,𝑧,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦,𝑧)   𝜃(𝑦,𝑧)

Proof of Theorem euabsn2w
StepHypRef Expression
1 euabsn2w.z . . 3 (𝑥 = 𝑧 → (𝜑𝜃))
2 absnw.y . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2eu6w 42686 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
41absnw 42688 . . 3 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝜑𝑥 = 𝑦))
54exbii 1848 . 2 (∃𝑦{𝑥𝜑} = {𝑦} ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
63, 5bitr4i 278 1 (∃!𝑥𝜑 ↔ ∃𝑦{𝑥𝜑} = {𝑦})
Colors of variables: wff setvar class
Syntax hints:  wb 206  wal 1538   = wceq 1540  wex 1779  ∃!weu 2568  {cab 2714  {csn 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-sn 4627
This theorem is referenced by:  sn-tz6.12-2  42690
  Copyright terms: Public domain W3C validator