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Theorem euabsn2w 42665
Description: Replace ax-10 2138, ax-11 2154, ax-12 2174 in euabsn2 4729 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 42662 . 2 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
41absnw 42664 . . 3 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝜑𝑥 = 𝑦))
54exbii 1844 . 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 1534   = wceq 1536  wex 1775  ∃!weu 2565  {cab 2711  {csn 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-sn 4631
This theorem is referenced by:  sn-tz6.12-2  42666
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