Theorem abbi1sn 41346

Description: Originally part of uniabio 6510. Convert a theorem about df-iota 6495 to one about dfiota2 6496, without ax-10 2137, ax-11 2154, ax-12 2171. Although, eu6 2568 uses ax-10 2137 and ax-12 2171. (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 2800	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦})
2		df-sn 4629	. 2 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦}
3	1, 2	eqtr4di 2790	1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦})

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1539   = wceq 1541  {cab 2709  {csn 4628

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 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-sn 4629

This theorem is referenced by: (None)