Theorem eliotaeu 5303

Description: An inhabited iota expression has a unique value. (Contributed by Jim Kingdon, 22-Nov-2024.)

Assertion

Ref	Expression
eliotaeu	⊢ (𝐴 ∈ (℩𝑥𝜑) → ∃!𝑥𝜑)

Proof of Theorem eliotaeu

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exsimpr 1664	. 2 ⊢ (∃𝑦(𝐴 ∈ 𝑦 ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2		eliota 5302	. 2 ⊢ (𝐴 ∈ (℩𝑥𝜑) ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
3		df-eu 2080	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
4	1, 2, 3	3imtr4i 201	1 ⊢ (𝐴 ∈ (℩𝑥𝜑) → ∃!𝑥𝜑)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1393  ∃wex 1538  ∃!weu 2077   ∈ wcel 2200  ℩cio 5272

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-sn 3672  df-uni 3888  df-iota 5274

This theorem is referenced by:  iotam  5306  elfvm  5656