Theorem abbi1sn 42854

Description: Originally part of uniabio 6495. Convert a theorem about df-iota 6481 to one about dfiota2 6482, without ax-10 2178, ax-11 2194, ax-12 2215. Although, eu6 2604 uses ax-10 2178 and ax-12 2215. (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 2830	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦})
2		df-sn 4586	. 2 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦}
3	1, 2	eqtr4di 2818	1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦})

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1561   = wceq 1563  {cab 2743  {csn 4585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-sn 4586

This theorem is referenced by: (None)