Theorem abbi1sn 40500

Description: Originally part of uniabio 6447. Convert a theorem about df-iota 6432 to one about dfiota2 6433, without ax-10 2136, ax-11 2153, ax-12 2170. Although, eu6 2572 uses ax-10 2136 and ax-12 2170. (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		abbi1 2804	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦})
2		df-sn 4575	. 2 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦}
3	1, 2	eqtr4di 2794	1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦})

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1538   = wceq 1540  {cab 2713  {csn 4574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-sn 4575

This theorem is referenced by: (None)