Theorem riotaxfrd 7442

Description: Change the variable 𝑥 in the expression for "the unique 𝑥 such that 𝜓 " to another variable 𝑦 contained in expression 𝐵. Use reuhypd 5388 to eliminate the last hypothesis. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
riotaxfrd.1	⊢ Ⅎ𝑦𝐶
riotaxfrd.2	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐵 ∈ 𝐴)
riotaxfrd.3	⊢ ((𝜑 ∧ (℩𝑦 ∈ 𝐴 𝜒) ∈ 𝐴) → 𝐶 ∈ 𝐴)
riotaxfrd.4	⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))
riotaxfrd.5	⊢ (𝑦 = (℩𝑦 ∈ 𝐴 𝜒) → 𝐵 = 𝐶)
riotaxfrd.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦 ∈ 𝐴 𝑥 = 𝐵)

Assertion

Ref	Expression
riotaxfrd	⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (℩𝑥 ∈ 𝐴 𝜓) = 𝐶)

Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝑦,𝐴   𝜑,𝑥,𝑦   𝜓,𝑦   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐵(𝑦)   𝐶(𝑦)

Proof of Theorem riotaxfrd

Step	Hyp	Ref	Expression
1		rabid 3436	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))
2	1	baib 545	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ 𝜓))
3	2	riotabiia 7428	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}) = (℩𝑥 ∈ 𝐴 𝜓)
4		riotaxfrd.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐵 ∈ 𝐴)
5		riotaxfrd.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦 ∈ 𝐴 𝑥 = 𝐵)
6		riotaxfrd.4	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))
7	4, 5, 6	reuxfr1ds 3712	. . . . 5 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑦 ∈ 𝐴 𝜒))
8		riotacl2 7424	. . . . . . . 8 ⊢ (∃!𝑦 ∈ 𝐴 𝜒 → (℩𝑦 ∈ 𝐴 𝜒) ∈ {𝑦 ∈ 𝐴 ∣ 𝜒})
9	8	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ ∃!𝑦 ∈ 𝐴 𝜒) → (℩𝑦 ∈ 𝐴 𝜒) ∈ {𝑦 ∈ 𝐴 ∣ 𝜒})
10		riotacl 7425	. . . . . . . 8 ⊢ (∃!𝑦 ∈ 𝐴 𝜒 → (℩𝑦 ∈ 𝐴 𝜒) ∈ 𝐴)
11		nfriota1 7415	. . . . . . . . 9 ⊢ Ⅎ𝑦(℩𝑦 ∈ 𝐴 𝜒)
12		riotaxfrd.1	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐶
13		riotaxfrd.5	. . . . . . . . 9 ⊢ (𝑦 = (℩𝑦 ∈ 𝐴 𝜒) → 𝐵 = 𝐶)
14	11, 12, 4, 6, 13	rabxfrd 5386	. . . . . . . 8 ⊢ ((𝜑 ∧ (℩𝑦 ∈ 𝐴 𝜒) ∈ 𝐴) → (𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (℩𝑦 ∈ 𝐴 𝜒) ∈ {𝑦 ∈ 𝐴 ∣ 𝜒}))
15	10, 14	sylan2 605	. . . . . . 7 ⊢ ((𝜑 ∧ ∃!𝑦 ∈ 𝐴 𝜒) → (𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (℩𝑦 ∈ 𝐴 𝜒) ∈ {𝑦 ∈ 𝐴 ∣ 𝜒}))
16	9, 15	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ ∃!𝑦 ∈ 𝐴 𝜒) → 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})
17	16	ex 418	. . . . 5 ⊢ (𝜑 → (∃!𝑦 ∈ 𝐴 𝜒 → 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}))
18	7, 17	sylbid 243	. . . 4 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 → 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}))
19	18	imp 412	. . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})
20		riotaxfrd.3	. . . . . . . 8 ⊢ ((𝜑 ∧ (℩𝑦 ∈ 𝐴 𝜒) ∈ 𝐴) → 𝐶 ∈ 𝐴)
21	20	ex 418	. . . . . . 7 ⊢ (𝜑 → ((℩𝑦 ∈ 𝐴 𝜒) ∈ 𝐴 → 𝐶 ∈ 𝐴))
22	10, 21	syl5 35	. . . . . 6 ⊢ (𝜑 → (∃!𝑦 ∈ 𝐴 𝜒 → 𝐶 ∈ 𝐴))
23	7, 22	sylbid 243	. . . . 5 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 → 𝐶 ∈ 𝐴))
24	23	imp 412	. . . 4 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → 𝐶 ∈ 𝐴)
25	1	baibr 546	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}))
26	25	reubiia 3376	. . . . 5 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})
27	26	bilani 510	. . . 4 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → ∃!𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})
28		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑥𝐶
29		nfrab1 3435	. . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜓}
30	29	nfel2 2944	. . . . 5 ⊢ Ⅎ𝑥 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}
31		eleq1 2852	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}))
32	28, 30, 31	riota2f 7432	. . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}) → (𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (℩𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}) = 𝐶))
33	24, 27, 32	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (℩𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}) = 𝐶))
34	19, 33	mpbid 235	. 2 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (℩𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}) = 𝐶)
35	3, 34	eqtr3id 2812	1 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (℩𝑥 ∈ 𝐴 𝜓) = 𝐶)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Ⅎwnfc 2911  ∃!wreu 3367  {crab 3416  ℩crio 7407

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-un 4154  df-sn 4589  df-pr 4591  df-uni 4873  df-iota 6520  df-riota 7408

This theorem is used by:  riotaneg  12336  zriotaneg  12852  riotaocN  40532