Theorem absnw 43128

Description: Replace ax-10 2152, ax-11 2168, ax-12 2189 in absn 4575 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 4556	. . 3 ⊢ {𝑌} = {𝑥 ∣ 𝑥 = 𝑌}
2	1	eqeq2i 2752	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑌} ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑌})
3		absnw.y	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4		eqeq1 2743	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑌 ↔ 𝑦 = 𝑌))
5	3, 4	abbibw 43127	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑌} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑌))
6	2, 5	bitri 276	1 ⊢ ({𝑥 ∣ 𝜑} = {𝑌} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑌))

Syntax hints:   ↔ wb 207  ∀wal 1545   = wceq 1547  {cab 2717  {csn 4555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-sn 4556

This theorem is referenced by:  euabsn2w  43129