Theorem abbi1sn 43022

Description: Originally part of uniabio 6506. Convert a theorem about df-iota 6492 to one about dfiota2 6493, without ax-10 2175, ax-11 2191, ax-12 2212. Although, eu6 2601 uses ax-10 2175 and ax-12 2212. (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 2827	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦})
2		df-sn 4589	. 2 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦}
3	1, 2	eqtr4di 2815	1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567   = wceq 1569  {cab 2740  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-sn 4589

This theorem is used by: (None)