Theorem abbi1sn 42218

Description: Originally part of uniabio 6542. Convert a theorem about df-iota 6527 to one about dfiota2 6528, without ax-10 2141, ax-11 2158, ax-12 2178. Although, eu6 2577 uses ax-10 2141 and ax-12 2178. (Contributed by SN, 23-Nov-2024.)

Assertion

Ref	Expression
abbi1sn	⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦})

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem abbi1sn

Step	Hyp	Ref	Expression
1		abbi 2810	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦})
2		df-sn 4649	. 2 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦}
3	1, 2	eqtr4di 2798	1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦})

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535   = wceq 1537  {cab 2717  {csn 4648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-sn 4649

This theorem is referenced by: (None)