Theorem iotain 37640

Description: Equivalence between two different forms of ℩. (Contributed by Andrew Salmon, 15-Jul-2011.)

Assertion

Ref	Expression
iotain	⊢ (∃!𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑))

Proof of Theorem iotain

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 2462	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2		vex 3176	. . . . 5 ⊢ 𝑦 ∈ V
3	2	intsn 4448	. . . 4 ⊢ ∩ {𝑦} = 𝑦
4		nfa1 2015	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥(𝜑 ↔ 𝑥 = 𝑦)
5		sp 2041	. . . . . . 7 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (𝜑 ↔ 𝑥 = 𝑦))
6	4, 5	abbid 2727	. . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦})
7		df-sn 4126	. . . . . 6 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦}
8	6, 7	syl6eqr 2662	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦})
9	8	inteqd 4415	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = ∩ {𝑦})
10		iotaval 5779	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
11	3, 9, 10	3eqtr4a 2670	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑))
12	11	exlimiv 1845	. 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑))
13	1, 12	sylbi 206	1 ⊢ (∃!𝑥𝜑 → ∩ {𝑥 ∣ 𝜑} = (℩𝑥𝜑))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475  ∃wex 1695  ∃!weu 2458  {cab 2596  {csn 4125  ∩ cint 4410  ℩cio 5766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-un 3545  df-in 3547  df-sn 4126  df-pr 4128  df-uni 4373  df-int 4411  df-iota 5768

This theorem is referenced by: (None)