Theorem iotaint 5325

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 5324	. 2 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑})
2		uniintabim 3985	. 2 ⊢ (∃!𝑥𝜑 → ∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑})
3	1, 2	eqtrd 2265	1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∩ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1398  ∃!weu 2080  {cab 2218  ∪ cuni 3913  ∩ cint 3948  ℩cio 5309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-sbc 3042  df-un 3214  df-in 3216  df-sn 3694  df-pr 3695  df-uni 3914  df-int 3949  df-iota 5311

This theorem is referenced by:  bdcriota  16640