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Theorem csbriota 6895
Description: Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 2-Sep-2018.)
Assertion
Ref Expression
csbriota 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑)
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem csbriota
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3754 . . . 4 (𝑧 = 𝐴𝑧 / 𝑥(𝑦𝐵 𝜑) = 𝐴 / 𝑥(𝑦𝐵 𝜑))
2 dfsbcq2 3655 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32riotabidv 6885 . . . 4 (𝑧 = 𝐴 → (𝑦𝐵 [𝑧 / 𝑥]𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
41, 3eqeq12d 2793 . . 3 (𝑧 = 𝐴 → (𝑧 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝑧 / 𝑥]𝜑) ↔ 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑)))
5 vex 3401 . . . 4 𝑧 ∈ V
6 nfs1v 2254 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
7 nfcv 2934 . . . . 5 𝑥𝐵
86, 7nfriota 6892 . . . 4 𝑥(𝑦𝐵 [𝑧 / 𝑥]𝜑)
9 sbequ12 2229 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
109riotabidv 6885 . . . 4 (𝑥 = 𝑧 → (𝑦𝐵 𝜑) = (𝑦𝐵 [𝑧 / 𝑥]𝜑))
115, 8, 10csbief 3776 . . 3 𝑧 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝑧 / 𝑥]𝜑)
124, 11vtoclg 3467 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
13 csbprc 4206 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝑦𝐵 𝜑) = ∅)
14 df-riota 6883 . . . 4 (𝑦𝐵 [𝐴 / 𝑥]𝜑) = (℩𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑))
15 euex 2597 . . . . . . 7 (∃!𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑) → ∃𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑))
16 sbcex 3662 . . . . . . . . 9 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
1716adantl 475 . . . . . . . 8 ((𝑦𝐵[𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
1817exlimiv 1973 . . . . . . 7 (∃𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
1915, 18syl 17 . . . . . 6 (∃!𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
2019con3i 152 . . . . 5 𝐴 ∈ V → ¬ ∃!𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑))
21 iotanul 6114 . . . . 5 (¬ ∃!𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑) → (℩𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑)) = ∅)
2220, 21syl 17 . . . 4 𝐴 ∈ V → (℩𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑)) = ∅)
2314, 22syl5req 2827 . . 3 𝐴 ∈ V → ∅ = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
2413, 23eqtrd 2814 . 2 𝐴 ∈ V → 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
2512, 24pm2.61i 177 1 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 386   = wceq 1601  wex 1823  [wsb 2011  wcel 2107  ∃!weu 2586  Vcvv 3398  [wsbc 3652  csb 3751  c0 4141  cio 6097  crio 6882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-in 3799  df-ss 3806  df-nul 4142  df-sn 4399  df-uni 4672  df-iota 6099  df-riota 6883
This theorem is referenced by:  cdlemkid3N  37087  cdlemkid4  37088
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