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Theorem riotaxfrd 7341
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 5373 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 3426 . . . 4 (𝑥 ∈ {𝑥𝐴𝜓} ↔ (𝑥𝐴𝜓))
21baib 537 . . 3 (𝑥𝐴 → (𝑥 ∈ {𝑥𝐴𝜓} ↔ 𝜓))
32riotabiia 7327 . 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, 6reuxfr1ds 3708 . . . . 5 (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑦𝐴 𝜒))
8 riotacl2 7323 . . . . . . . 8 (∃!𝑦𝐴 𝜒 → (𝑦𝐴 𝜒) ∈ {𝑦𝐴𝜒})
98adantl 483 . . . . . . 7 ((𝜑 ∧ ∃!𝑦𝐴 𝜒) → (𝑦𝐴 𝜒) ∈ {𝑦𝐴𝜒})
10 riotacl 7324 . . . . . . . 8 (∃!𝑦𝐴 𝜒 → (𝑦𝐴 𝜒) ∈ 𝐴)
11 nfriota1 7313 . . . . . . . . 9 𝑦(𝑦𝐴 𝜒)
12 riotaxfrd.1 . . . . . . . . 9 𝑦𝐶
13 riotaxfrd.5 . . . . . . . . 9 (𝑦 = (𝑦𝐴 𝜒) → 𝐵 = 𝐶)
1411, 12, 4, 6, 13rabxfrd 5371 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴 𝜒) ∈ 𝐴) → (𝐶 ∈ {𝑥𝐴𝜓} ↔ (𝑦𝐴 𝜒) ∈ {𝑦𝐴𝜒}))
1510, 14sylan2 594 . . . . . . 7 ((𝜑 ∧ ∃!𝑦𝐴 𝜒) → (𝐶 ∈ {𝑥𝐴𝜓} ↔ (𝑦𝐴 𝜒) ∈ {𝑦𝐴𝜒}))
169, 15mpbird 257 . . . . . 6 ((𝜑 ∧ ∃!𝑦𝐴 𝜒) → 𝐶 ∈ {𝑥𝐴𝜓})
1716ex 414 . . . . 5 (𝜑 → (∃!𝑦𝐴 𝜒𝐶 ∈ {𝑥𝐴𝜓}))
187, 17sylbid 239 . . . 4 (𝜑 → (∃!𝑥𝐴 𝜓𝐶 ∈ {𝑥𝐴𝜓}))
1918imp 408 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → 𝐶 ∈ {𝑥𝐴𝜓})
20 riotaxfrd.3 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴 𝜒) ∈ 𝐴) → 𝐶𝐴)
2120ex 414 . . . . . . 7 (𝜑 → ((𝑦𝐴 𝜒) ∈ 𝐴𝐶𝐴))
2210, 21syl5 34 . . . . . 6 (𝜑 → (∃!𝑦𝐴 𝜒𝐶𝐴))
237, 22sylbid 239 . . . . 5 (𝜑 → (∃!𝑥𝐴 𝜓𝐶𝐴))
2423imp 408 . . . 4 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → 𝐶𝐴)
251baibr 538 . . . . . . 7 (𝑥𝐴 → (𝜓𝑥 ∈ {𝑥𝐴𝜓}))
2625reubiia 3359 . . . . . 6 (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓})
2726biimpi 215 . . . . 5 (∃!𝑥𝐴 𝜓 → ∃!𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓})
2827adantl 483 . . . 4 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → ∃!𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓})
29 nfcv 2906 . . . . 5 𝑥𝐶
30 nfrab1 3425 . . . . . 6 𝑥{𝑥𝐴𝜓}
3130nfel2 2924 . . . . 5 𝑥 𝐶 ∈ {𝑥𝐴𝜓}
32 eleq1 2826 . . . . 5 (𝑥 = 𝐶 → (𝑥 ∈ {𝑥𝐴𝜓} ↔ 𝐶 ∈ {𝑥𝐴𝜓}))
3329, 31, 32riota2f 7331 . . . 4 ((𝐶𝐴 ∧ ∃!𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓}) → (𝐶 ∈ {𝑥𝐴𝜓} ↔ (𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓}) = 𝐶))
3424, 28, 33syl2anc 585 . . 3 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝐶 ∈ {𝑥𝐴𝜓} ↔ (𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓}) = 𝐶))
3519, 34mpbid 231 . 2 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝑥𝐴 𝑥 ∈ {𝑥𝐴𝜓}) = 𝐶)
363, 35eqtr3id 2792 1 ((𝜑 ∧ ∃!𝑥𝐴 𝜓) → (𝑥𝐴 𝜓) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wnfc 2886  ∃!wreu 3350  {crab 3406  crio 7305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-un 3914  df-in 3916  df-ss 3926  df-sn 4586  df-pr 4588  df-uni 4865  df-iota 6444  df-riota 7306
This theorem is referenced by:  riotaneg  12068  zriotaneg  12550  riotaocN  37602
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