Theorem absnw 42238

Description: Replace ax-10 2129, ax-11 2146, ax-12 2166 in absn 4649 with a substitution hypothesis. (Contributed by SN, 27-May-2025.)

Hypothesis

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

Assertion

Ref	Expression
absnw	⊢ ({𝑥 ∣ 𝜑} = {𝑌} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑌))

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦,𝑌

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem absnw

Step	Hyp	Ref	Expression
1		df-sn 4631	. . 3 ⊢ {𝑌} = {𝑥 ∣ 𝑥 = 𝑌}
2	1	eqeq2i 2738	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑌} ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑌})
3		absnw.y	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4		eqeq1 2729	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑌 ↔ 𝑦 = 𝑌))
5	3, 4	abbibw 42237	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑌} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑌))
6	2, 5	bitri 274	1 ⊢ ({𝑥 ∣ 𝜑} = {𝑌} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑌))

Syntax hints:   ↔ wb 205  ∀wal 1531   = wceq 1533  {cab 2702  {csn 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-sn 4631

This theorem is referenced by:  euabsn2w  42239