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Theorem riotaeqimp 6798
 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 2769 . . . . . 6 (𝑎𝑉 𝑌 = 𝐴) = 𝐽
32eqeq2i 2772 . . . . 5 (𝐼 = (𝑎𝑉 𝑌 = 𝐴) ↔ 𝐼 = 𝐽)
43a1i 11 . . . 4 (𝜑 → (𝐼 = (𝑎𝑉 𝑌 = 𝐴) ↔ 𝐼 = 𝐽))
54bicomd 213 . . 3 (𝜑 → (𝐼 = 𝐽𝐼 = (𝑎𝑉 𝑌 = 𝐴)))
65biimpa 502 . 2 ((𝜑𝐼 = 𝐽) → 𝐼 = (𝑎𝑉 𝑌 = 𝐴))
7 riotaeqimp.i . . . . 5 𝐼 = (𝑎𝑉 𝑋 = 𝐴)
87eqeq1i 2765 . . . 4 (𝐼 = 𝐽 ↔ (𝑎𝑉 𝑋 = 𝐴) = 𝐽)
9 riotaeqimp.y . . . . . . 7 (𝜑 → ∃!𝑎𝑉 𝑌 = 𝐴)
10 riotacl 6789 . . . . . . 7 (∃!𝑎𝑉 𝑌 = 𝐴 → (𝑎𝑉 𝑌 = 𝐴) ∈ 𝑉)
119, 10syl 17 . . . . . 6 (𝜑 → (𝑎𝑉 𝑌 = 𝐴) ∈ 𝑉)
121, 11syl5eqel 2843 . . . . 5 (𝜑𝐽𝑉)
13 riotaeqimp.x . . . . 5 (𝜑 → ∃!𝑎𝑉 𝑋 = 𝐴)
14 nfv 1992 . . . . . . 7 𝑎 𝐽𝑉
15 nfcvd 2903 . . . . . . 7 (𝐽𝑉𝑎𝐽)
16 nfcvd 2903 . . . . . . . 8 (𝐽𝑉𝑎𝑋)
1715nfcsb1d 3688 . . . . . . . 8 (𝐽𝑉𝑎𝐽 / 𝑎𝐴)
1816, 17nfeqd 2910 . . . . . . 7 (𝐽𝑉 → Ⅎ𝑎 𝑋 = 𝐽 / 𝑎𝐴)
19 id 22 . . . . . . 7 (𝐽𝑉𝐽𝑉)
20 csbeq1a 3683 . . . . . . . . 9 (𝑎 = 𝐽𝐴 = 𝐽 / 𝑎𝐴)
2120eqeq2d 2770 . . . . . . . 8 (𝑎 = 𝐽 → (𝑋 = 𝐴𝑋 = 𝐽 / 𝑎𝐴))
2221adantl 473 . . . . . . 7 ((𝐽𝑉𝑎 = 𝐽) → (𝑋 = 𝐴𝑋 = 𝐽 / 𝑎𝐴))
2314, 15, 18, 19, 22riota2df 6795 . . . . . 6 ((𝐽𝑉 ∧ ∃!𝑎𝑉 𝑋 = 𝐴) → (𝑋 = 𝐽 / 𝑎𝐴 ↔ (𝑎𝑉 𝑋 = 𝐴) = 𝐽))
2423bicomd 213 . . . . 5 ((𝐽𝑉 ∧ ∃!𝑎𝑉 𝑋 = 𝐴) → ((𝑎𝑉 𝑋 = 𝐴) = 𝐽𝑋 = 𝐽 / 𝑎𝐴))
2512, 13, 24syl2anc 696 . . . 4 (𝜑 → ((𝑎𝑉 𝑋 = 𝐴) = 𝐽𝑋 = 𝐽 / 𝑎𝐴))
268, 25syl5bb 272 . . 3 (𝜑 → (𝐼 = 𝐽𝑋 = 𝐽 / 𝑎𝐴))
2726biimpa 502 . 2 ((𝜑𝐼 = 𝐽) → 𝑋 = 𝐽 / 𝑎𝐴)
28 riotacl 6789 . . . . . . . 8 (∃!𝑎𝑉 𝑋 = 𝐴 → (𝑎𝑉 𝑋 = 𝐴) ∈ 𝑉)
2913, 28syl 17 . . . . . . 7 (𝜑 → (𝑎𝑉 𝑋 = 𝐴) ∈ 𝑉)
307, 29syl5eqel 2843 . . . . . 6 (𝜑𝐼𝑉)
31 nfv 1992 . . . . . . 7 𝑎 𝐼𝑉
32 nfcvd 2903 . . . . . . 7 (𝐼𝑉𝑎𝐼)
33 nfcvd 2903 . . . . . . . 8 (𝐼𝑉𝑎𝑌)
3432nfcsb1d 3688 . . . . . . . 8 (𝐼𝑉𝑎𝐼 / 𝑎𝐴)
3533, 34nfeqd 2910 . . . . . . 7 (𝐼𝑉 → Ⅎ𝑎 𝑌 = 𝐼 / 𝑎𝐴)
36 id 22 . . . . . . 7 (𝐼𝑉𝐼𝑉)
37 csbeq1a 3683 . . . . . . . . 9 (𝑎 = 𝐼𝐴 = 𝐼 / 𝑎𝐴)
3837eqeq2d 2770 . . . . . . . 8 (𝑎 = 𝐼 → (𝑌 = 𝐴𝑌 = 𝐼 / 𝑎𝐴))
3938adantl 473 . . . . . . 7 ((𝐼𝑉𝑎 = 𝐼) → (𝑌 = 𝐴𝑌 = 𝐼 / 𝑎𝐴))
4031, 32, 35, 36, 39riota2df 6795 . . . . . 6 ((𝐼𝑉 ∧ ∃!𝑎𝑉 𝑌 = 𝐴) → (𝑌 = 𝐼 / 𝑎𝐴 ↔ (𝑎𝑉 𝑌 = 𝐴) = 𝐼))
4130, 9, 40syl2anc 696 . . . . 5 (𝜑 → (𝑌 = 𝐼 / 𝑎𝐴 ↔ (𝑎𝑉 𝑌 = 𝐴) = 𝐼))
42 eqcom 2767 . . . . 5 ((𝑎𝑉 𝑌 = 𝐴) = 𝐼𝐼 = (𝑎𝑉 𝑌 = 𝐴))
4341, 42syl6bb 276 . . . 4 (𝜑 → (𝑌 = 𝐼 / 𝑎𝐴𝐼 = (𝑎𝑉 𝑌 = 𝐴)))
4443adantr 472 . . 3 ((𝜑𝐼 = 𝐽) → (𝑌 = 𝐼 / 𝑎𝐴𝐼 = (𝑎𝑉 𝑌 = 𝐴)))
45 csbeq1 3677 . . . . . . 7 (𝐽 = 𝐼𝐽 / 𝑎𝐴 = 𝐼 / 𝑎𝐴)
4645eqcoms 2768 . . . . . 6 (𝐼 = 𝐽𝐽 / 𝑎𝐴 = 𝐼 / 𝑎𝐴)
47 eqeq12 2773 . . . . . . 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 237 . . . . 5 (𝐼 = 𝐽 → ((𝑌 = 𝐼 / 𝑎𝐴𝑋 = 𝐽 / 𝑎𝐴) → 𝑋 = 𝑌))
5049expd 451 . . . 4 (𝐼 = 𝐽 → (𝑌 = 𝐼 / 𝑎𝐴 → (𝑋 = 𝐽 / 𝑎𝐴𝑋 = 𝑌)))
5150adantl 473 . . 3 ((𝜑𝐼 = 𝐽) → (𝑌 = 𝐼 / 𝑎𝐴 → (𝑋 = 𝐽 / 𝑎𝐴𝑋 = 𝑌)))
5244, 51sylbird 250 . 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 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∃!wreu 3052  ⦋csb 3674  ℩crio 6774 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-un 3720  df-in 3722  df-ss 3729  df-sn 4322  df-pr 4324  df-uni 4589  df-iota 6012  df-riota 6775 This theorem is referenced by:  uspgredg2v  26336
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