Theorem iotaexeu 44774

Description: The iota class exists. This theorem does not require ax-nul 5253 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 6474	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
2	1	eqcomd 2743	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
3	2	eximi 1837	. 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦 𝑦 = (℩𝑥𝜑))
4		eu6 2575	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
5		isset 3456	. 2 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃𝑦 𝑦 = (℩𝑥𝜑))
6	3, 4, 5	3imtr4i 292	1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃!weu 2569  Vcvv 3442  ℩cio 6454

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-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-un 3908  df-ss 3920  df-sn 4583  df-pr 4585  df-uni 4866  df-iota 6456

This theorem is referenced by:  iotasbc  44775  pm14.18  44784  iotavalb  44786  sbiota1  44790