Theorem iota2 6552

Description: The unique element such that 𝜑. (Contributed by Jeff Madsen, 1-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Hypothesis

Ref	Expression
iota2.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
iota2	⊢ ((𝐴 ∈ 𝐵 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem iota2

Step	Hyp	Ref	Expression
1		elex 3499	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
2		simpl 482	. . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → 𝐴 ∈ V)
3		simpr 484	. . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → ∃!𝑥𝜑)
4		iota2.1	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	4	adantl 481	. . 3 ⊢ (((𝐴 ∈ V ∧ ∃!𝑥𝜑) ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓))
6		nfv 1912	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V
7		nfeu1 2586	. . . 4 ⊢ Ⅎ𝑥∃!𝑥𝜑
8	6, 7	nfan 1897	. . 3 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ ∃!𝑥𝜑)
9		nfvd 1913	. . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → Ⅎ𝑥𝜓)
10		nfcvd 2904	. . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → Ⅎ𝑥𝐴)
11	2, 3, 5, 8, 9, 10	iota2df 6550	. 2 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
12	1, 11	sylan 580	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∃!weu 2566  Vcvv 3478  ℩cio 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634  df-uni 4913  df-iota 6516

This theorem is referenced by:  iotan0  6553  pczpre  16881  pcdiv  16886  ringurd  20203  nosupno  27763  nosupfv  27766  noinfno  27778  noinffv  27781  bj-nuliota  37040  unirep  37701  ellimciota  45570