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Theorem riotaxfrd 6597
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 4860 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 3111 . . . 4 (𝑥 ∈ {𝑥𝐴𝜓} ↔ (𝑥𝐴𝜓))
21baib 943 . . 3 (𝑥𝐴 → (𝑥 ∈ {𝑥𝐴𝜓} ↔ 𝜓))
32riotabiia 6583 . 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 4858 . . . . 5 (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑦𝐴 𝜒))
8 riotacl2 6579 . . . . . . . 8 (∃!𝑦𝐴 𝜒 → (𝑦𝐴 𝜒) ∈ {𝑦𝐴𝜒})
98adantl 482 . . . . . . 7 ((𝜑 ∧ ∃!𝑦𝐴 𝜒) → (𝑦𝐴 𝜒) ∈ {𝑦𝐴𝜒})
10 riotacl 6580 . . . . . . . 8 (∃!𝑦𝐴 𝜒 → (𝑦𝐴 𝜒) ∈ 𝐴)
11 nfriota1 6573 . . . . . . . . 9 𝑦(𝑦𝐴 𝜒)
12 riotaxfrd.1 . . . . . . . . 9 𝑦𝐶
13 riotaxfrd.5 . . . . . . . . 9 (𝑦 = (𝑦𝐴 𝜒) → 𝐵 = 𝐶)
1411, 12, 4, 6, 13rabxfrd 4854 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴 𝜒) ∈ 𝐴) → (𝐶 ∈ {𝑥𝐴𝜓} ↔ (𝑦𝐴 𝜒) ∈ {𝑦𝐴𝜒}))
1510, 14sylan2 491 . . . . . . 7 ((𝜑 ∧ ∃!𝑦𝐴 𝜒) → (𝐶 ∈ {𝑥𝐴𝜓} ↔ (𝑦𝐴 𝜒) ∈ {𝑦𝐴𝜒}))
169, 15mpbird 247 . . . . . 6 ((𝜑 ∧ ∃!𝑦𝐴 𝜒) → 𝐶 ∈ {𝑥𝐴𝜓})
1716ex 450 . . . . 5 (𝜑 → (∃!𝑦𝐴 𝜒𝐶 ∈ {𝑥𝐴𝜓}))
187, 17sylbid 230 . . . 4 (𝜑 → (∃!𝑥𝐴 𝜓𝐶 ∈ {𝑥𝐴𝜓}))
1918imp 445 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → 𝐶 ∈ {𝑥𝐴𝜓})
20 riotaxfrd.3 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴 𝜒) ∈ 𝐴) → 𝐶𝐴)
2120ex 450 . . . . . . 7 (𝜑 → ((𝑦𝐴 𝜒) ∈ 𝐴𝐶𝐴))
2210, 21syl5 34 . . . . . 6 (𝜑 → (∃!𝑦𝐴 𝜒𝐶𝐴))
237, 22sylbid 230 . . . . 5 (𝜑 → (∃!𝑥𝐴 𝜓𝐶𝐴))
2423imp 445 . . . 4 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → 𝐶𝐴)
251baibr 944 . . . . . . 7 (𝑥𝐴 → (𝜓𝑥 ∈ {𝑥𝐴𝜓}))
2625reubiia 3121 . . . . . 6 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓})
2726biimpi 206 . . . . 5 (∃!𝑥𝐴 𝜓 → ∃!𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓})
2827adantl 482 . . . 4 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → ∃!𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓})
29 nfcv 2767 . . . . 5 𝑥𝐶
30 nfrab1 3116 . . . . . 6 𝑥{𝑥𝐴𝜓}
3130nfel2 2783 . . . . 5 𝑥 𝐶 ∈ {𝑥𝐴𝜓}
32 eleq1 2692 . . . . 5 (𝑥 = 𝐶 → (𝑥 ∈ {𝑥𝐴𝜓} ↔ 𝐶 ∈ {𝑥𝐴𝜓}))
3329, 31, 32riota2f 6587 . . . 4 ((𝐶𝐴 ∧ ∃!𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓}) → (𝐶 ∈ {𝑥𝐴𝜓} ↔ (𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓}) = 𝐶))
3424, 28, 33syl2anc 692 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝐶 ∈ {𝑥𝐴𝜓} ↔ (𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓}) = 𝐶))
3519, 34mpbid 222 . 2 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓}) = 𝐶)
363, 35syl5eqr 2674 1 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝑥𝐴 𝜓) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wnfc 2754  ∃!wreu 2914  {crab 2916  crio 6565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-un 3565  df-in 3567  df-ss 3574  df-sn 4154  df-pr 4156  df-uni 4408  df-iota 5813  df-riota 6566
This theorem is referenced by:  riotaneg  10947  zriotaneg  11435  riotaocN  33962
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