Theorem sn-iotaval 40192

Description: iotaval 6406 without ax-10 2141, ax-11 2158, ax-12 2175. (Contributed by SN, 23-Nov-2024.)

Assertion

Ref	Expression
sn-iotaval	⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem sn-iotaval

Step	Hyp	Ref	Expression
1		abbi1sn 40188	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦})
2		iotavallem 40189	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
3	1, 2	syl 17	1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1540   = wceq 1542  {cab 2717  {csn 4567  ℩cio 6388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-v 3433  df-un 3897  df-in 3899  df-ss 3909  df-sn 4568  df-pr 4570  df-uni 4846  df-iota 6390

This theorem is referenced by: (None)