Theorem euabsn2w 42660

Description: Replace ax-10 2142, ax-11 2158, ax-12 2178 in euabsn2 4685 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 42657	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
4	1	absnw 42659	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
5	4	exbii 1848	. 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
6	3, 5	bitr4i 278	1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦})

Syntax hints:   ↔ wb 206  ∀wal 1538   = wceq 1540  ∃wex 1779  ∃!weu 2561  {cab 2707  {csn 4585

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3446  df-sn 4586

This theorem is referenced by:  sn-tz6.12-2  42661