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Theorem riotaeqimp 7120
 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 2807 . . . . . 6 (𝑎𝑉 𝑌 = 𝐴) = 𝐽
32eqeq2i 2811 . . . . 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 2803 . . . 4 (𝐼 = 𝐽 ↔ (𝑎𝑉 𝑋 = 𝐴) = 𝐽)
9 riotaeqimp.y . . . . . . 7 (𝜑 → ∃!𝑎𝑉 𝑌 = 𝐴)
10 riotacl 7111 . . . . . . 7 (∃!𝑎𝑉 𝑌 = 𝐴 → (𝑎𝑉 𝑌 = 𝐴) ∈ 𝑉)
119, 10syl 17 . . . . . 6 (𝜑 → (𝑎𝑉 𝑌 = 𝐴) ∈ 𝑉)
121, 11eqeltrid 2894 . . . . 5 (𝜑𝐽𝑉)
13 riotaeqimp.x . . . . 5 (𝜑 → ∃!𝑎𝑉 𝑋 = 𝐴)
14 nfv 1915 . . . . . . 7 𝑎 𝐽𝑉
15 nfcvd 2956 . . . . . . 7 (𝐽𝑉𝑎𝐽)
16 nfcvd 2956 . . . . . . . 8 (𝐽𝑉𝑎𝑋)
1715nfcsb1d 3850 . . . . . . . 8 (𝐽𝑉𝑎𝐽 / 𝑎𝐴)
1816, 17nfeqd 2965 . . . . . . 7 (𝐽𝑉 → Ⅎ𝑎 𝑋 = 𝐽 / 𝑎𝐴)
19 id 22 . . . . . . 7 (𝐽𝑉𝐽𝑉)
20 csbeq1a 3842 . . . . . . . . 9 (𝑎 = 𝐽𝐴 = 𝐽 / 𝑎𝐴)
2120eqeq2d 2809 . . . . . . . 8 (𝑎 = 𝐽 → (𝑋 = 𝐴𝑋 = 𝐽 / 𝑎𝐴))
2221adantl 485 . . . . . . 7 ((𝐽𝑉𝑎 = 𝐽) → (𝑋 = 𝐴𝑋 = 𝐽 / 𝑎𝐴))
2314, 15, 18, 19, 22riota2df 7117 . . . . . 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 7111 . . . . . . . 8 (∃!𝑎𝑉 𝑋 = 𝐴 → (𝑎𝑉 𝑋 = 𝐴) ∈ 𝑉)
2913, 28syl 17 . . . . . . 7 (𝜑 → (𝑎𝑉 𝑋 = 𝐴) ∈ 𝑉)
307, 29eqeltrid 2894 . . . . . 6 (𝜑𝐼𝑉)
31 nfv 1915 . . . . . . 7 𝑎 𝐼𝑉
32 nfcvd 2956 . . . . . . 7 (𝐼𝑉𝑎𝐼)
33 nfcvd 2956 . . . . . . . 8 (𝐼𝑉𝑎𝑌)
3432nfcsb1d 3850 . . . . . . . 8 (𝐼𝑉𝑎𝐼 / 𝑎𝐴)
3533, 34nfeqd 2965 . . . . . . 7 (𝐼𝑉 → Ⅎ𝑎 𝑌 = 𝐼 / 𝑎𝐴)
36 id 22 . . . . . . 7 (𝐼𝑉𝐼𝑉)
37 csbeq1a 3842 . . . . . . . . 9 (𝑎 = 𝐼𝐴 = 𝐼 / 𝑎𝐴)
3837eqeq2d 2809 . . . . . . . 8 (𝑎 = 𝐼 → (𝑌 = 𝐴𝑌 = 𝐼 / 𝑎𝐴))
3938adantl 485 . . . . . . 7 ((𝐼𝑉𝑎 = 𝐼) → (𝑌 = 𝐴𝑌 = 𝐼 / 𝑎𝐴))
4031, 32, 35, 36, 39riota2df 7117 . . . . . 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 2805 . . . . 5 ((𝑎𝑉 𝑌 = 𝐴) = 𝐼𝐼 = (𝑎𝑉 𝑌 = 𝐴))
4341, 42syl6bb 290 . . . 4 (𝜑 → (𝑌 = 𝐼 / 𝑎𝐴𝐼 = (𝑎𝑉 𝑌 = 𝐴)))
4443adantr 484 . . 3 ((𝜑𝐼 = 𝐽) → (𝑌 = 𝐼 / 𝑎𝐴𝐼 = (𝑎𝑉 𝑌 = 𝐴)))
45 csbeq1 3831 . . . . . . 7 (𝐽 = 𝐼𝐽 / 𝑎𝐴 = 𝐼 / 𝑎𝐴)
4645eqcoms 2806 . . . . . 6 (𝐼 = 𝐽𝐽 / 𝑎𝐴 = 𝐼 / 𝑎𝐴)
47 eqeq12 2812 . . . . . . 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 1538   ∈ wcel 2111  ∃!wreu 3108  ⦋csb 3828  ℩crio 7093 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-uni 4802  df-iota 6284  df-riota 7094 This theorem is referenced by:  uspgredg2v  27024
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