Theorem iotaint 6464

Description: Equivalence between two different forms of ℩. (Contributed by Mario Carneiro, 24-Dec-2016.)

Assertion

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

Proof of Theorem iotaint

Step	Hyp	Ref	Expression
1		iotauni 6463	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
2		uniintab 4936	. . 3 ⊢ (∃!𝑥𝜑 ↔ ∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑})
3	2	biimpi 216	. 2 ⊢ (∃!𝑥𝜑 → ∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑})
4	1, 3	eqtrd 2768	1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1541  ∃!weu 2565  {cab 2711  ∪ cuni 4858  ∩ cint 4897  ℩cio 6440

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-sn 4576  df-pr 4578  df-uni 4859  df-int 4898  df-iota 6442

This theorem is referenced by:  aiotaint  47215