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Theorem riotaxfrd 6797
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 5036 to eliminate the last hypothesis. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
riotaxfrd.1 𝑦𝐶
riotaxfrd.2 ((𝜑𝑦𝐴) → 𝐵𝐴)
riotaxfrd.3 ((𝜑 ∧ (𝑦𝐴 𝜒) ∈ 𝐴) → 𝐶𝐴)
riotaxfrd.4 (𝑥 = 𝐵 → (𝜓𝜒))
riotaxfrd.5 (𝑦 = (𝑦𝐴 𝜒) → 𝐵 = 𝐶)
riotaxfrd.6 ((𝜑𝑥𝐴) → ∃!𝑦𝐴 𝑥 = 𝐵)
Assertion
Ref Expression
riotaxfrd ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝑥𝐴 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
StepHypRef Expression
1 rabid 3246 . . . 4 (𝑥 ∈ {𝑥𝐴𝜓} ↔ (𝑥𝐴𝜓))
21baib 982 . . 3 (𝑥𝐴 → (𝑥 ∈ {𝑥𝐴𝜓} ↔ 𝜓))
32riotabiia 6783 . 2 (𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓}) = (𝑥𝐴 𝜓)
4 riotaxfrd.2 . . . . . 6 ((𝜑𝑦𝐴) → 𝐵𝐴)
5 riotaxfrd.6 . . . . . 6 ((𝜑𝑥𝐴) → ∃!𝑦𝐴 𝑥 = 𝐵)
6 riotaxfrd.4 . . . . . 6 (𝑥 = 𝐵 → (𝜓𝜒))
74, 5, 6reuxfrd 5034 . . . . 5 (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑦𝐴 𝜒))
8 riotacl2 6779 . . . . . . . 8 (∃!𝑦𝐴 𝜒 → (𝑦𝐴 𝜒) ∈ {𝑦𝐴𝜒})
98adantl 473 . . . . . . 7 ((𝜑 ∧ ∃!𝑦𝐴 𝜒) → (𝑦𝐴 𝜒) ∈ {𝑦𝐴𝜒})
10 riotacl 6780 . . . . . . . 8 (∃!𝑦𝐴 𝜒 → (𝑦𝐴 𝜒) ∈ 𝐴)
11 nfriota1 6773 . . . . . . . . 9 𝑦(𝑦𝐴 𝜒)
12 riotaxfrd.1 . . . . . . . . 9 𝑦𝐶
13 riotaxfrd.5 . . . . . . . . 9 (𝑦 = (𝑦𝐴 𝜒) → 𝐵 = 𝐶)
1411, 12, 4, 6, 13rabxfrd 5030 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴 𝜒) ∈ 𝐴) → (𝐶 ∈ {𝑥𝐴𝜓} ↔ (𝑦𝐴 𝜒) ∈ {𝑦𝐴𝜒}))
1510, 14sylan2 492 . . . . . . 7 ((𝜑 ∧ ∃!𝑦𝐴 𝜒) → (𝐶 ∈ {𝑥𝐴𝜓} ↔ (𝑦𝐴 𝜒) ∈ {𝑦𝐴𝜒}))
169, 15mpbird 247 . . . . . 6 ((𝜑 ∧ ∃!𝑦𝐴 𝜒) → 𝐶 ∈ {𝑥𝐴𝜓})
1716ex 449 . . . . 5 (𝜑 → (∃!𝑦𝐴 𝜒𝐶 ∈ {𝑥𝐴𝜓}))
187, 17sylbid 230 . . . 4 (𝜑 → (∃!𝑥𝐴 𝜓𝐶 ∈ {𝑥𝐴𝜓}))
1918imp 444 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → 𝐶 ∈ {𝑥𝐴𝜓})
20 riotaxfrd.3 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴 𝜒) ∈ 𝐴) → 𝐶𝐴)
2120ex 449 . . . . . . 7 (𝜑 → ((𝑦𝐴 𝜒) ∈ 𝐴𝐶𝐴))
2210, 21syl5 34 . . . . . 6 (𝜑 → (∃!𝑦𝐴 𝜒𝐶𝐴))
237, 22sylbid 230 . . . . 5 (𝜑 → (∃!𝑥𝐴 𝜓𝐶𝐴))
2423imp 444 . . . 4 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → 𝐶𝐴)
251baibr 983 . . . . . . 7 (𝑥𝐴 → (𝜓𝑥 ∈ {𝑥𝐴𝜓}))
2625reubiia 3258 . . . . . 6 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓})
2726biimpi 206 . . . . 5 (∃!𝑥𝐴 𝜓 → ∃!𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓})
2827adantl 473 . . . 4 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → ∃!𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓})
29 nfcv 2894 . . . . 5 𝑥𝐶
30 nfrab1 3253 . . . . . 6 𝑥{𝑥𝐴𝜓}
3130nfel2 2911 . . . . 5 𝑥 𝐶 ∈ {𝑥𝐴𝜓}
32 eleq1 2819 . . . . 5 (𝑥 = 𝐶 → (𝑥 ∈ {𝑥𝐴𝜓} ↔ 𝐶 ∈ {𝑥𝐴𝜓}))
3329, 31, 32riota2f 6787 . . . 4 ((𝐶𝐴 ∧ ∃!𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓}) → (𝐶 ∈ {𝑥𝐴𝜓} ↔ (𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓}) = 𝐶))
3424, 28, 33syl2anc 696 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝐶 ∈ {𝑥𝐴𝜓} ↔ (𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓}) = 𝐶))
3519, 34mpbid 222 . 2 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓}) = 𝐶)
363, 35syl5eqr 2800 1 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝑥𝐴 𝜓) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  wnfc 2881  ∃!wreu 3044  {crab 3046  crio 6765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-un 3712  df-in 3714  df-ss 3721  df-sn 4314  df-pr 4316  df-uni 4581  df-iota 6004  df-riota 6766
This theorem is referenced by:  riotaneg  11186  zriotaneg  11675  riotaocN  34991
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