Theorem aiotaexaiotaiota 47984

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

Assertion

Ref	Expression
aiotaexaiotaiota	⊢ ((℩'𝑥𝜑) ∈ V ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))

Proof of Theorem aiotaexaiotaiota

Step	Hyp	Ref	Expression
1		aiotaexb 47979	. 2 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
2		reuaiotaiota 47978	. 2 ⊢ (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
3	1, 2	bitr3i 280	1 ⊢ ((℩'𝑥𝜑) ∈ V ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∃!weu 2595  Vcvv 3453  ℩cio 6491  ℩'caiota 47973

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-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-sn 4588  df-pr 4590  df-uni 4871  df-int 4911  df-iota 6493  df-aiota 47975

This theorem is used by: (None)