Theorem riotaxfrd 6541

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 4821 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 3095	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (𝑥 ∈ 𝐴 ∧ 𝜓))
2	1	baib 942	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ 𝜓))
3	2	riotabiia 6528	. 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	reuxfrd 4819	. . . . 5 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑦 ∈ 𝐴 𝜒))
8		riotacl2 6524	. . . . . . . 8 ⊢ (∃!𝑦 ∈ 𝐴 𝜒 → (℩𝑦 ∈ 𝐴 𝜒) ∈ {𝑦 ∈ 𝐴 ∣ 𝜒})
9	8	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ ∃!𝑦 ∈ 𝐴 𝜒) → (℩𝑦 ∈ 𝐴 𝜒) ∈ {𝑦 ∈ 𝐴 ∣ 𝜒})
10		riotacl 6525	. . . . . . . 8 ⊢ (∃!𝑦 ∈ 𝐴 𝜒 → (℩𝑦 ∈ 𝐴 𝜒) ∈ 𝐴)
11		nfriota1 6518	. . . . . . . . 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 4815	. . . . . . . 8 ⊢ ((𝜑 ∧ (℩𝑦 ∈ 𝐴 𝜒) ∈ 𝐴) → (𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (℩𝑦 ∈ 𝐴 𝜒) ∈ {𝑦 ∈ 𝐴 ∣ 𝜒}))
15	10, 14	sylan2 490	. . . . . . 7 ⊢ ((𝜑 ∧ ∃!𝑦 ∈ 𝐴 𝜒) → (𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (℩𝑦 ∈ 𝐴 𝜒) ∈ {𝑦 ∈ 𝐴 ∣ 𝜒}))
16	9, 15	mpbird 246	. . . . . 6 ⊢ ((𝜑 ∧ ∃!𝑦 ∈ 𝐴 𝜒) → 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})
17	16	ex 449	. . . . 5 ⊢ (𝜑 → (∃!𝑦 ∈ 𝐴 𝜒 → 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}))
18	7, 17	sylbid 229	. . . 4 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 → 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}))
19	18	imp 444	. . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})
20		riotaxfrd.3	. . . . . . . 8 ⊢ ((𝜑 ∧ (℩𝑦 ∈ 𝐴 𝜒) ∈ 𝐴) → 𝐶 ∈ 𝐴)
21	20	ex 449	. . . . . . 7 ⊢ (𝜑 → ((℩𝑦 ∈ 𝐴 𝜒) ∈ 𝐴 → 𝐶 ∈ 𝐴))
22	10, 21	syl5 33	. . . . . 6 ⊢ (𝜑 → (∃!𝑦 ∈ 𝐴 𝜒 → 𝐶 ∈ 𝐴))
23	7, 22	sylbid 229	. . . . 5 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 → 𝐶 ∈ 𝐴))
24	23	imp 444	. . . 4 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → 𝐶 ∈ 𝐴)
25	1	baibr 943	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝜓 ↔ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}))
26	25	reubiia 3104	. . . . . 6 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})
27	26	biimpi 205	. . . . 5 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 → ∃!𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})
28	27	adantl 481	. . . 4 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → ∃!𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓})
29		nfcv 2751	. . . . 5 ⊢ Ⅎ𝑥𝐶
30		nfrab1 3099	. . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜓}
31	30	nfel2 2767	. . . . 5 ⊢ Ⅎ𝑥 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}
32		eleq1 2676	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ 𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}))
33	29, 31, 32	riota2f 6532	. . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃!𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}) → (𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (℩𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}) = 𝐶))
34	24, 28, 33	syl2anc 691	. . 3 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (𝐶 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓} ↔ (℩𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}) = 𝐶))
35	19, 34	mpbid 221	. 2 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (℩𝑥 ∈ 𝐴 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜓}) = 𝐶)
36	3, 35	syl5eqr 2658	1 ⊢ ((𝜑 ∧ ∃!𝑥 ∈ 𝐴 𝜓) → (℩𝑥 ∈ 𝐴 𝜓) = 𝐶)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  ∃!wreu 2898  {crab 2900  ℩crio 6510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-un 3545  df-in 3547  df-ss 3554  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768  df-riota 6511

This theorem is referenced by:  riotaneg  10879  zriotaneg  11367  riotaocN  33514