Theorem iota2d 6499

Description: A condition that allows to represent "the unique element such that 𝜑 " with a class expression 𝐴. (Contributed by NM, 30-Dec-2014.)

Hypotheses

Ref	Expression
iota2df.1	⊢ (𝜑 → 𝐵 ∈ 𝑉)
iota2df.2	⊢ (𝜑 → ∃!𝑥𝜓)
iota2df.3	⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
iota2d	⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))

Distinct variable groups:   𝑥,𝐵   𝜑,𝑥   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem iota2d

Step	Hyp	Ref	Expression
1		iota2df.1	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
2		iota2df.2	. 2 ⊢ (𝜑 → ∃!𝑥𝜓)
3		iota2df.3	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒))
4		nfv 1914	. 2 ⊢ Ⅎ𝑥𝜑
5		nfvd 1915	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
6		nfcvd 2892	. 2 ⊢ (𝜑 → Ⅎ𝑥𝐵)
7	1, 2, 3, 4, 5, 6	iota2df 6498	1 ⊢ (𝜑 → (𝜒 ↔ (℩𝑥𝜓) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃!weu 2561  ℩cio 6462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-v 3449  df-un 3919  df-ss 3931  df-sn 4590  df-pr 4592  df-uni 4872  df-iota 6464

This theorem is referenced by:  erov  8787  psgnvalii  19439  q1peqb  26061