Theorem reuaiotaiota 47193

Description: The iota and the alternate iota over a wff 𝜑 are equal iff there is a unique value 𝑥 satisfying 𝜑. (Contributed by AV, 25-Aug-2022.)

Assertion

Ref	Expression
reuaiotaiota	⊢ (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))

Proof of Theorem reuaiotaiota

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		euabsneu 47133	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦{𝑥 ∣ 𝜑} = {𝑦})
2		reuabaiotaiota 47192	. 2 ⊢ (∃!𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
3	1, 2	bitri 275	1 ⊢ (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))

Syntax hints:   ↔ wb 206   = wceq 1541  ∃!weu 2563  {cab 2709  {csn 4575  ℩cio 6441  ℩'caiota 47188

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-sn 4576  df-pr 4578  df-uni 4859  df-int 4898  df-iota 6443  df-aiota 47190

This theorem is referenced by:  aiotaint  47196  aiotaexaiotaiota  47199  dfafv2  47237