Theorem iotaexeu 40770

Description: The iota class exists. This theorem does not require ax-nul 5210 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 6329	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
2	1	eqcomd 2827	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
3	2	eximi 1835	. 2 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦 𝑦 = (℩𝑥𝜑))
4		eu6 2659	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
5		isset 3506	. 2 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃𝑦 𝑦 = (℩𝑥𝜑))
6	3, 4, 5	3imtr4i 294	1 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∃!weu 2653  Vcvv 3494  ℩cio 6312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-sbc 3773  df-un 3941  df-in 3943  df-ss 3952  df-sn 4568  df-pr 4570  df-uni 4839  df-iota 6314

This theorem is referenced by:  iotasbc  40771  pm14.18  40780  iotavalb  40782  sbiota1  40786