Theorem riota2df 7385

Description: A deduction version of riota2f 7386. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
riota2df.1	⊢ Ⅎ𝑥𝜑
riota2df.2	⊢ (𝜑 → Ⅎ𝑥𝐵)
riota2df.3	⊢ (𝜑 → Ⅎ𝑥𝜒)
riota2df.4	⊢ (𝜑 → 𝐵 ∈ 𝐴)
riota2df.5	⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
riota2df	⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (𝜒 ↔ (℩𝑥 ∈ 𝐴 𝜓) = 𝐵))

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem riota2df

Step	Hyp	Ref	Expression
1		riota2df.4	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
2	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → 𝐵 ∈ 𝐴)
3		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → ∃!𝑥 ∈ 𝐴 𝜓)
4		df-reu 3371	. . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
5	3, 4	sylib 217	. . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
6		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
7	2	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝐵 ∈ 𝐴)
8	6, 7	eqeltrd 2827	. . . . 5 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝑥 ∈ 𝐴)
9	8	biantrurd 532	. . . 4 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → (𝜓 ↔ (𝑥 ∈ 𝐴 ∧ 𝜓)))
10		riota2df.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒))
11	10	adantlr 712	. . . 4 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒))
12	9, 11	bitr3d 281	. . 3 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ 𝜒))
13		riota2df.1	. . . 4 ⊢ Ⅎ𝑥𝜑
14		nfreu1 3402	. . . 4 ⊢ Ⅎ𝑥∃!𝑥 ∈ 𝐴 𝜓
15	13, 14	nfan 1894	. . 3 ⊢ Ⅎ𝑥(𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓)
16		riota2df.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒)
17	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → Ⅎ𝑥𝜒)
18		riota2df.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵)
19	18	adantr 480	. . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → Ⅎ𝑥𝐵)
20	2, 5, 12, 15, 17, 19	iota2df 6524	. 2 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (𝜒 ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = 𝐵))
21		df-riota 7361	. . 3 ⊢ (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
22	21	eqeq1i 2731	. 2 ⊢ ((℩𝑥 ∈ 𝐴 𝜓) = 𝐵 ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = 𝐵)
23	20, 22	bitr4di 289	1 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (𝜒 ↔ (℩𝑥 ∈ 𝐴 𝜓) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533  Ⅎwnf 1777   ∈ wcel 2098  ∃!weu 2556  Ⅎwnfc 2877  ∃!wreu 3368  ℩cio 6487  ℩crio 7360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-reu 3371  df-v 3470  df-un 3948  df-in 3950  df-ss 3960  df-sn 4624  df-pr 4626  df-uni 4903  df-iota 6489  df-riota 7361

This theorem is referenced by:  riota2f  7386  riotaeqimp  7388  riota5f  7390  mapdheq  41112  hdmap1eq  41185  hdmapval2lem  41215