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Theorem riotaeqimp 40338
 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 2619 . . . . . 6 (𝑎𝑉 𝑌 = 𝐴) = 𝐽
32eqeq2i 2622 . . . . 5 (𝐼 = (𝑎𝑉 𝑌 = 𝐴) ↔ 𝐼 = 𝐽)
43a1i 11 . . . 4 (𝜑 → (𝐼 = (𝑎𝑉 𝑌 = 𝐴) ↔ 𝐼 = 𝐽))
54bicomd 212 . . 3 (𝜑 → (𝐼 = 𝐽𝐼 = (𝑎𝑉 𝑌 = 𝐴)))
65biimpa 500 . 2 ((𝜑𝐼 = 𝐽) → 𝐼 = (𝑎𝑉 𝑌 = 𝐴))
7 riotaeqimp.i . . . . 5 𝐼 = (𝑎𝑉 𝑋 = 𝐴)
87eqeq1i 2615 . . . 4 (𝐼 = 𝐽 ↔ (𝑎𝑉 𝑋 = 𝐴) = 𝐽)
9 riotaeqimp.y . . . . . . 7 (𝜑 → ∃!𝑎𝑉 𝑌 = 𝐴)
10 riotacl 6525 . . . . . . 7 (∃!𝑎𝑉 𝑌 = 𝐴 → (𝑎𝑉 𝑌 = 𝐴) ∈ 𝑉)
119, 10syl 17 . . . . . 6 (𝜑 → (𝑎𝑉 𝑌 = 𝐴) ∈ 𝑉)
121, 11syl5eqel 2692 . . . . 5 (𝜑𝐽𝑉)
13 riotaeqimp.x . . . . 5 (𝜑 → ∃!𝑎𝑉 𝑋 = 𝐴)
14 nfv 1830 . . . . . . 7 𝑎 𝐽𝑉
15 nfcvd 2752 . . . . . . 7 (𝐽𝑉𝑎𝐽)
16 nfcvd 2752 . . . . . . . 8 (𝐽𝑉𝑎𝑋)
1715nfcsb1d 3513 . . . . . . . 8 (𝐽𝑉𝑎𝐽 / 𝑎𝐴)
1816, 17nfeqd 2758 . . . . . . 7 (𝐽𝑉 → Ⅎ𝑎 𝑋 = 𝐽 / 𝑎𝐴)
19 id 22 . . . . . . 7 (𝐽𝑉𝐽𝑉)
20 csbeq1a 3508 . . . . . . . . 9 (𝑎 = 𝐽𝐴 = 𝐽 / 𝑎𝐴)
2120eqeq2d 2620 . . . . . . . 8 (𝑎 = 𝐽 → (𝑋 = 𝐴𝑋 = 𝐽 / 𝑎𝐴))
2221adantl 481 . . . . . . 7 ((𝐽𝑉𝑎 = 𝐽) → (𝑋 = 𝐴𝑋 = 𝐽 / 𝑎𝐴))
2314, 15, 18, 19, 22riota2df 6531 . . . . . 6 ((𝐽𝑉 ∧ ∃!𝑎𝑉 𝑋 = 𝐴) → (𝑋 = 𝐽 / 𝑎𝐴 ↔ (𝑎𝑉 𝑋 = 𝐴) = 𝐽))
2423bicomd 212 . . . . 5 ((𝐽𝑉 ∧ ∃!𝑎𝑉 𝑋 = 𝐴) → ((𝑎𝑉 𝑋 = 𝐴) = 𝐽𝑋 = 𝐽 / 𝑎𝐴))
2512, 13, 24syl2anc 691 . . . 4 (𝜑 → ((𝑎𝑉 𝑋 = 𝐴) = 𝐽𝑋 = 𝐽 / 𝑎𝐴))
268, 25syl5bb 271 . . 3 (𝜑 → (𝐼 = 𝐽𝑋 = 𝐽 / 𝑎𝐴))
2726biimpa 500 . 2 ((𝜑𝐼 = 𝐽) → 𝑋 = 𝐽 / 𝑎𝐴)
28 riotacl 6525 . . . . . . . 8 (∃!𝑎𝑉 𝑋 = 𝐴 → (𝑎𝑉 𝑋 = 𝐴) ∈ 𝑉)
2913, 28syl 17 . . . . . . 7 (𝜑 → (𝑎𝑉 𝑋 = 𝐴) ∈ 𝑉)
307, 29syl5eqel 2692 . . . . . 6 (𝜑𝐼𝑉)
31 nfv 1830 . . . . . . 7 𝑎 𝐼𝑉
32 nfcvd 2752 . . . . . . 7 (𝐼𝑉𝑎𝐼)
33 nfcvd 2752 . . . . . . . 8 (𝐼𝑉𝑎𝑌)
3432nfcsb1d 3513 . . . . . . . 8 (𝐼𝑉𝑎𝐼 / 𝑎𝐴)
3533, 34nfeqd 2758 . . . . . . 7 (𝐼𝑉 → Ⅎ𝑎 𝑌 = 𝐼 / 𝑎𝐴)
36 id 22 . . . . . . 7 (𝐼𝑉𝐼𝑉)
37 csbeq1a 3508 . . . . . . . . 9 (𝑎 = 𝐼𝐴 = 𝐼 / 𝑎𝐴)
3837eqeq2d 2620 . . . . . . . 8 (𝑎 = 𝐼 → (𝑌 = 𝐴𝑌 = 𝐼 / 𝑎𝐴))
3938adantl 481 . . . . . . 7 ((𝐼𝑉𝑎 = 𝐼) → (𝑌 = 𝐴𝑌 = 𝐼 / 𝑎𝐴))
4031, 32, 35, 36, 39riota2df 6531 . . . . . 6 ((𝐼𝑉 ∧ ∃!𝑎𝑉 𝑌 = 𝐴) → (𝑌 = 𝐼 / 𝑎𝐴 ↔ (𝑎𝑉 𝑌 = 𝐴) = 𝐼))
4130, 9, 40syl2anc 691 . . . . 5 (𝜑 → (𝑌 = 𝐼 / 𝑎𝐴 ↔ (𝑎𝑉 𝑌 = 𝐴) = 𝐼))
42 eqcom 2617 . . . . 5 ((𝑎𝑉 𝑌 = 𝐴) = 𝐼𝐼 = (𝑎𝑉 𝑌 = 𝐴))
4341, 42syl6bb 275 . . . 4 (𝜑 → (𝑌 = 𝐼 / 𝑎𝐴𝐼 = (𝑎𝑉 𝑌 = 𝐴)))
4443adantr 480 . . 3 ((𝜑𝐼 = 𝐽) → (𝑌 = 𝐼 / 𝑎𝐴𝐼 = (𝑎𝑉 𝑌 = 𝐴)))
45 csbeq1 3502 . . . . . . 7 (𝐽 = 𝐼𝐽 / 𝑎𝐴 = 𝐼 / 𝑎𝐴)
4645eqcoms 2618 . . . . . 6 (𝐼 = 𝐽𝐽 / 𝑎𝐴 = 𝐼 / 𝑎𝐴)
47 eqeq12 2623 . . . . . . 7 ((𝑋 = 𝐽 / 𝑎𝐴𝑌 = 𝐼 / 𝑎𝐴) → (𝑋 = 𝑌𝐽 / 𝑎𝐴 = 𝐼 / 𝑎𝐴))
4847ancoms 468 . . . . . 6 ((𝑌 = 𝐼 / 𝑎𝐴𝑋 = 𝐽 / 𝑎𝐴) → (𝑋 = 𝑌𝐽 / 𝑎𝐴 = 𝐼 / 𝑎𝐴))
4946, 48syl5ibrcom 236 . . . . 5 (𝐼 = 𝐽 → ((𝑌 = 𝐼 / 𝑎𝐴𝑋 = 𝐽 / 𝑎𝐴) → 𝑋 = 𝑌))
5049expd 451 . . . 4 (𝐼 = 𝐽 → (𝑌 = 𝐼 / 𝑎𝐴 → (𝑋 = 𝐽 / 𝑎𝐴𝑋 = 𝑌)))
5150adantl 481 . . 3 ((𝜑𝐼 = 𝐽) → (𝑌 = 𝐼 / 𝑎𝐴 → (𝑋 = 𝐽 / 𝑎𝐴𝑋 = 𝑌)))
5244, 51sylbird 249 . 2 ((𝜑𝐼 = 𝐽) → (𝐼 = (𝑎𝑉 𝑌 = 𝐴) → (𝑋 = 𝐽 / 𝑎𝐴𝑋 = 𝑌)))
536, 27, 52mp2d 47 1 ((𝜑𝐼 = 𝐽) → 𝑋 = 𝑌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃!wreu 2898  ⦋csb 3499  ℩crio 6510 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-un 3545  df-in 3547  df-ss 3554  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768  df-riota 6511 This theorem is referenced by:  uspgredg2v  40451
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