Theorem euabsn2w 43221

Description: Replace ax-10 2174, ax-11 2190, ax-12 2211 in euabsn2 4681 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 43218	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
4	1	absnw 43220	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
5	4	exbii 1867	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
6	3, 5	bitr4i 280	1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})

Syntax hints:   ↔ wb 208  ∀wal 1557   = wceq 1559  ∃wex 1798  ∃!weu 2594  {cab 2739  {csn 4579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-sn 4580

This theorem is referenced by: (None)