Theorem euabsn2w 42958

Description: Replace ax-10 2147, ax-11 2163, ax-12 2185 in euabsn2 4683 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 42955	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
4	1	absnw 42957	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
5	4	exbii 1850	. 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 1540   = wceq 1542  ∃wex 1781  ∃!weu 2569  {cab 2715  {csn 4581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3443  df-sn 4582

This theorem is referenced by: (None)