Theorem iotauni2 6509

Description: Version of iotauni 6514 using df-iota 6493 instead of dfiota2 6494. (Contributed by SN, 6-Nov-2024.)

Assertion

Ref	Expression
iotauni2	⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem iotauni2

Step	Hyp	Ref	Expression
1		iotaval2 6508	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = 𝑦)
2		unieq 4881	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → ∪ {𝑥 ∣ 𝜑} = ∪ {𝑦})
3		unisnv 4890	. . . 4 ⊢ ∪ {𝑦} = 𝑦
4	2, 3	eqtr2di 2814	. . 3 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → 𝑦 = ∪ {𝑥 ∣ 𝜑})
5	1, 4	eqtrd 2797	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
6	5	exlimiv 1963	1 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ∃wex 1812  {cab 2740  {csn 4587  ∪ cuni 4870  ℩cio 6491

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-un 3907  df-ss 3919  df-sn 4588  df-pr 4590  df-uni 4871  df-iota 6493

This theorem is used by:  iotassuni  6512