Theorem euiotaex 5168

Description: Theorem 8.23 in [Quine] p. 58, with existential uniqueness condition added. This theorem proves the existence of the ℩ class under our definition. (Contributed by Jim Kingdon, 21-Dec-2018.)

Assertion

Ref	Expression
euiotaex	⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)

Proof of Theorem euiotaex

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iotaval 5163	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
2	1	eqcomd 2171	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
3	2	eximi 1588	. 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦 𝑦 = (℩𝑥𝜑))
4		df-eu 2017	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
5		isset 2731	. 2 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃𝑦 𝑦 = (℩𝑥𝜑))
6	3, 4, 5	3imtr4i 200	1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1341   = wceq 1343  ∃wex 1480  ∃!weu 2014   ∈ wcel 2136  Vcvv 2725  ℩cio 5150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-rex 2449  df-v 2727  df-sbc 2951  df-un 3119  df-sn 3581  df-pr 3582  df-uni 3789  df-iota 5152

This theorem is referenced by:  iota4an  5171  funfvex  5502  pcval  12224