Theorem aiotaexaiotaiota 47542

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 47537	. 2 ⊢ (∃!𝑥𝜑 ↔ (℩'𝑥𝜑) ∈ V)
2		reuaiotaiota 47536	. 2 ⊢ (∃!𝑥𝜑 ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))
3	1, 2	bitr3i 277	1 ⊢ ((℩'𝑥𝜑) ∈ V ↔ (℩𝑥𝜑) = (℩'𝑥𝜑))

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  ∃!weu 2568  Vcvv 3429  ℩cio 6452  ℩'caiota 47531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-sn 4568  df-pr 4570  df-uni 4851  df-int 4890  df-iota 6454  df-aiota 47533

This theorem is referenced by: (None)