Theorem riota2df 7340

Description: A deduction version of riota2f 7341. (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 482	. . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → 𝐵 ∈ 𝐴)
3		df-reu 3347	. . . 4 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
4	3	bilani 506	. . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
5		simpr 486	. . . . . 6 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
6	2	adantr 482	. . . . . 6 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝐵 ∈ 𝐴)
7	5, 6	eqeltrd 2841	. . . . 5 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → 𝑥 ∈ 𝐴)
8	7	biantrurd 538	. . . 4 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → (𝜓 ↔ (𝑥 ∈ 𝐴 ∧ 𝜓)))
9		riota2df.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒))
10	9	adantlr 722	. . . 4 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → (𝜓 ↔ 𝜒))
11	8, 10	bitr3d 283	. . 3 ⊢ (((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 = 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ 𝜒))
12		riota2df.1	. . . 4 ⊢ Ⅎ𝑥𝜑
13		nfreu1 3374	. . . 4 ⊢ Ⅎ𝑥∃!𝑥 ∈ 𝐴 𝜓
14	12, 13	nfan 1907	. . 3 ⊢ Ⅎ𝑥(𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓)
15		riota2df.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒)
16	15	adantr 482	. . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → Ⅎ𝑥𝜒)
17		riota2df.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐵)
18	17	adantr 482	. . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → Ⅎ𝑥𝐵)
19	2, 4, 11, 14, 16, 18	iota2df 6476	. 2 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (𝜒 ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = 𝐵))
20		df-riota 7317	. . 3 ⊢ (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
21	20	eqeq1i 2746	. 2 ⊢ ((℩𝑥 ∈ 𝐴 𝜓) = 𝐵 ↔ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = 𝐵)
22	19, 21	bitr4di 291	1 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (𝜒 ↔ (℩𝑥 ∈ 𝐴 𝜓) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548  Ⅎwnf 1791   ∈ wcel 2121  ∃!weu 2574  Ⅎwnfc 2888  ∃!wreu 3344  ℩cio 6443  ℩crio 7316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056  df-rex 3066  df-reu 3347  df-v 3435  df-un 3890  df-ss 3902  df-sn 4559  df-pr 4561  df-uni 4842  df-iota 6445  df-riota 7317

This theorem is referenced by:  riota2f  7341  riotaeqimp  7343  riota5f  7345  mapdheq  42235  hdmap1eq  42308  hdmapval2lem  42338