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Theorem riotaeqimp 7197
Description: If two restricted iota descriptors for an equality are equal, then the terms of the equality are equal. (Contributed by AV, 6-Dec-2020.)
Hypotheses
Ref Expression
riotaeqimp.i 𝐼 = (𝑎𝑉 𝑋 = 𝐴)
riotaeqimp.j 𝐽 = (𝑎𝑉 𝑌 = 𝐴)
riotaeqimp.x (𝜑 → ∃!𝑎𝑉 𝑋 = 𝐴)
riotaeqimp.y (𝜑 → ∃!𝑎𝑉 𝑌 = 𝐴)
Assertion
Ref Expression
riotaeqimp ((𝜑𝐼 = 𝐽) → 𝑋 = 𝑌)
Distinct variable groups:   𝐼,𝑎   𝐽,𝑎   𝑉,𝑎   𝑋,𝑎   𝑌,𝑎
Allowed substitution hints:   𝜑(𝑎)   𝐴(𝑎)

Proof of Theorem riotaeqimp
StepHypRef Expression
1 riotaeqimp.j . . . . . . 7 𝐽 = (𝑎𝑉 𝑌 = 𝐴)
21eqcomi 2746 . . . . . 6 (𝑎𝑉 𝑌 = 𝐴) = 𝐽
32eqeq2i 2750 . . . . 5 (𝐼 = (𝑎𝑉 𝑌 = 𝐴) ↔ 𝐼 = 𝐽)
43a1i 11 . . . 4 (𝜑 → (𝐼 = (𝑎𝑉 𝑌 = 𝐴) ↔ 𝐼 = 𝐽))
54bicomd 226 . . 3 (𝜑 → (𝐼 = 𝐽𝐼 = (𝑎𝑉 𝑌 = 𝐴)))
65biimpa 480 . 2 ((𝜑𝐼 = 𝐽) → 𝐼 = (𝑎𝑉 𝑌 = 𝐴))
7 riotaeqimp.i . . . . 5 𝐼 = (𝑎𝑉 𝑋 = 𝐴)
87eqeq1i 2742 . . . 4 (𝐼 = 𝐽 ↔ (𝑎𝑉 𝑋 = 𝐴) = 𝐽)
9 riotaeqimp.y . . . . . . 7 (𝜑 → ∃!𝑎𝑉 𝑌 = 𝐴)
10 riotacl 7188 . . . . . . 7 (∃!𝑎𝑉 𝑌 = 𝐴 → (𝑎𝑉 𝑌 = 𝐴) ∈ 𝑉)
119, 10syl 17 . . . . . 6 (𝜑 → (𝑎𝑉 𝑌 = 𝐴) ∈ 𝑉)
121, 11eqeltrid 2842 . . . . 5 (𝜑𝐽𝑉)
13 riotaeqimp.x . . . . 5 (𝜑 → ∃!𝑎𝑉 𝑋 = 𝐴)
14 nfv 1922 . . . . . . 7 𝑎 𝐽𝑉
15 nfcvd 2905 . . . . . . 7 (𝐽𝑉𝑎𝐽)
16 nfcvd 2905 . . . . . . . 8 (𝐽𝑉𝑎𝑋)
1715nfcsb1d 3834 . . . . . . . 8 (𝐽𝑉𝑎𝐽 / 𝑎𝐴)
1816, 17nfeqd 2914 . . . . . . 7 (𝐽𝑉 → Ⅎ𝑎 𝑋 = 𝐽 / 𝑎𝐴)
19 id 22 . . . . . . 7 (𝐽𝑉𝐽𝑉)
20 csbeq1a 3825 . . . . . . . . 9 (𝑎 = 𝐽𝐴 = 𝐽 / 𝑎𝐴)
2120eqeq2d 2748 . . . . . . . 8 (𝑎 = 𝐽 → (𝑋 = 𝐴𝑋 = 𝐽 / 𝑎𝐴))
2221adantl 485 . . . . . . 7 ((𝐽𝑉𝑎 = 𝐽) → (𝑋 = 𝐴𝑋 = 𝐽 / 𝑎𝐴))
2314, 15, 18, 19, 22riota2df 7194 . . . . . 6 ((𝐽𝑉 ∧ ∃!𝑎𝑉 𝑋 = 𝐴) → (𝑋 = 𝐽 / 𝑎𝐴 ↔ (𝑎𝑉 𝑋 = 𝐴) = 𝐽))
2423bicomd 226 . . . . 5 ((𝐽𝑉 ∧ ∃!𝑎𝑉 𝑋 = 𝐴) → ((𝑎𝑉 𝑋 = 𝐴) = 𝐽𝑋 = 𝐽 / 𝑎𝐴))
2512, 13, 24syl2anc 587 . . . 4 (𝜑 → ((𝑎𝑉 𝑋 = 𝐴) = 𝐽𝑋 = 𝐽 / 𝑎𝐴))
268, 25syl5bb 286 . . 3 (𝜑 → (𝐼 = 𝐽𝑋 = 𝐽 / 𝑎𝐴))
2726biimpa 480 . 2 ((𝜑𝐼 = 𝐽) → 𝑋 = 𝐽 / 𝑎𝐴)
28 riotacl 7188 . . . . . . . 8 (∃!𝑎𝑉 𝑋 = 𝐴 → (𝑎𝑉 𝑋 = 𝐴) ∈ 𝑉)
2913, 28syl 17 . . . . . . 7 (𝜑 → (𝑎𝑉 𝑋 = 𝐴) ∈ 𝑉)
307, 29eqeltrid 2842 . . . . . 6 (𝜑𝐼𝑉)
31 nfv 1922 . . . . . . 7 𝑎 𝐼𝑉
32 nfcvd 2905 . . . . . . 7 (𝐼𝑉𝑎𝐼)
33 nfcvd 2905 . . . . . . . 8 (𝐼𝑉𝑎𝑌)
3432nfcsb1d 3834 . . . . . . . 8 (𝐼𝑉𝑎𝐼 / 𝑎𝐴)
3533, 34nfeqd 2914 . . . . . . 7 (𝐼𝑉 → Ⅎ𝑎 𝑌 = 𝐼 / 𝑎𝐴)
36 id 22 . . . . . . 7 (𝐼𝑉𝐼𝑉)
37 csbeq1a 3825 . . . . . . . . 9 (𝑎 = 𝐼𝐴 = 𝐼 / 𝑎𝐴)
3837eqeq2d 2748 . . . . . . . 8 (𝑎 = 𝐼 → (𝑌 = 𝐴𝑌 = 𝐼 / 𝑎𝐴))
3938adantl 485 . . . . . . 7 ((𝐼𝑉𝑎 = 𝐼) → (𝑌 = 𝐴𝑌 = 𝐼 / 𝑎𝐴))
4031, 32, 35, 36, 39riota2df 7194 . . . . . 6 ((𝐼𝑉 ∧ ∃!𝑎𝑉 𝑌 = 𝐴) → (𝑌 = 𝐼 / 𝑎𝐴 ↔ (𝑎𝑉 𝑌 = 𝐴) = 𝐼))
4130, 9, 40syl2anc 587 . . . . 5 (𝜑 → (𝑌 = 𝐼 / 𝑎𝐴 ↔ (𝑎𝑉 𝑌 = 𝐴) = 𝐼))
42 eqcom 2744 . . . . 5 ((𝑎𝑉 𝑌 = 𝐴) = 𝐼𝐼 = (𝑎𝑉 𝑌 = 𝐴))
4341, 42bitrdi 290 . . . 4 (𝜑 → (𝑌 = 𝐼 / 𝑎𝐴𝐼 = (𝑎𝑉 𝑌 = 𝐴)))
4443adantr 484 . . 3 ((𝜑𝐼 = 𝐽) → (𝑌 = 𝐼 / 𝑎𝐴𝐼 = (𝑎𝑉 𝑌 = 𝐴)))
45 csbeq1 3814 . . . . . . 7 (𝐽 = 𝐼𝐽 / 𝑎𝐴 = 𝐼 / 𝑎𝐴)
4645eqcoms 2745 . . . . . 6 (𝐼 = 𝐽𝐽 / 𝑎𝐴 = 𝐼 / 𝑎𝐴)
47 eqeq12 2754 . . . . . . 7 ((𝑋 = 𝐽 / 𝑎𝐴𝑌 = 𝐼 / 𝑎𝐴) → (𝑋 = 𝑌𝐽 / 𝑎𝐴 = 𝐼 / 𝑎𝐴))
4847ancoms 462 . . . . . 6 ((𝑌 = 𝐼 / 𝑎𝐴𝑋 = 𝐽 / 𝑎𝐴) → (𝑋 = 𝑌𝐽 / 𝑎𝐴 = 𝐼 / 𝑎𝐴))
4946, 48syl5ibrcom 250 . . . . 5 (𝐼 = 𝐽 → ((𝑌 = 𝐼 / 𝑎𝐴𝑋 = 𝐽 / 𝑎𝐴) → 𝑋 = 𝑌))
5049expd 419 . . . 4 (𝐼 = 𝐽 → (𝑌 = 𝐼 / 𝑎𝐴 → (𝑋 = 𝐽 / 𝑎𝐴𝑋 = 𝑌)))
5150adantl 485 . . 3 ((𝜑𝐼 = 𝐽) → (𝑌 = 𝐼 / 𝑎𝐴 → (𝑋 = 𝐽 / 𝑎𝐴𝑋 = 𝑌)))
5244, 51sylbird 263 . 2 ((𝜑𝐼 = 𝐽) → (𝐼 = (𝑎𝑉 𝑌 = 𝐴) → (𝑋 = 𝐽 / 𝑎𝐴𝑋 = 𝑌)))
536, 27, 52mp2d 49 1 ((𝜑𝐼 = 𝐽) → 𝑋 = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  ∃!wreu 3063  csb 3811  crio 7169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-un 3871  df-in 3873  df-ss 3883  df-sn 4542  df-pr 4544  df-uni 4820  df-iota 6338  df-riota 7170
This theorem is referenced by:  uspgredg2v  27312
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