Theorem iota2 5347

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 2827	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
2		simpl 109	. . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → 𝐴 ∈ V)
3		simpr 110	. . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → ∃!𝑥𝜑)
4		iota2.1	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	4	adantl 277	. . 3 ⊢ (((𝐴 ∈ V ∧ ∃!𝑥𝜑) ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓))
6		nfv 1577	. . . 4 ⊢ Ⅎ𝑥 𝐴 ∈ V
7		nfeu1 2093	. . . 4 ⊢ Ⅎ𝑥∃!𝑥𝜑
8	6, 7	nfan 1614	. . 3 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ ∃!𝑥𝜑)
9		nfvd 1578	. . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → Ⅎ𝑥𝜓)
10		nfcvd 2387	. . 3 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → Ⅎ𝑥𝐴)
11	2, 3, 5, 8, 9, 10	iota2df 5343	. 2 ⊢ ((𝐴 ∈ V ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))
12	1, 11	sylan 283	1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∃!𝑥𝜑) → (𝜓 ↔ (℩𝑥𝜑) = 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃!weu 2082   ∈ wcel 2205  Vcvv 2815  ℩cio 5315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-sn 3700  df-pr 3701  df-uni 3920  df-iota 5317

This theorem is referenced by:  iotam  5349  fvmbr  5710  pczpre  13020  pcdiv  13025  gsum0g  13659  gsumval2  13660  gfsumval  14102