Theorem euabsn2w 43956

Description: Replace ax-10 2180, ax-11 2197, ax-12 2216 in euabsn2 4688 with substitution hypotheses. (Contributed by SN, 27-May-2025.)

Hypotheses

Ref	Expression
absnw.y	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
euabsn2w.z	⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜃))

Assertion

Ref	Expression
euabsn2w	⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {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

Step	Hyp	Ref	Expression
1		euabsn2w.z	. . 3 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜃))
2		absnw.y	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	eu6w 43953	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
4	1	absnw 43955	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
5	4	exbii 1881	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
6	3, 5	bitr4i 281	1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1568   = wceq 1570  ∃wex 1812  ∃!weu 2596  {cab 2741  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-sn 4589

This theorem is used by: (None)