Theorem iotaexeu 43726

Description: The iota class exists. This theorem does not require ax-nul 5297 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

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

Proof of Theorem iotaexeu

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iotaval 6505	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
2	1	eqcomd 2730	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
3	2	eximi 1829	. 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦 𝑦 = (℩𝑥𝜑))
4		eu6 2560	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
5		isset 3479	. 2 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃𝑦 𝑦 = (℩𝑥𝜑))
6	3, 4, 5	3imtr4i 292	1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1531   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∃!weu 2554  Vcvv 3466  ℩cio 6484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-un 3946  df-in 3948  df-ss 3958  df-sn 4622  df-pr 4624  df-uni 4901  df-iota 6486

This theorem is referenced by:  iotasbc  43727  pm14.18  43736  iotavalb  43738  sbiota1  43742