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Theorem csbriota 7381
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 3897 . . . 4 (𝑧 = 𝐴𝑧 / 𝑥(𝑦𝐵 𝜑) = 𝐴 / 𝑥(𝑦𝐵 𝜑))
2 dfsbcq2 3781 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32riotabidv 7367 . . . 4 (𝑧 = 𝐴 → (𝑦𝐵 [𝑧 / 𝑥]𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
41, 3eqeq12d 2749 . . 3 (𝑧 = 𝐴 → (𝑧 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝑧 / 𝑥]𝜑) ↔ 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑)))
5 vex 3479 . . . 4 𝑧 ∈ V
6 nfs1v 2154 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
7 nfcv 2904 . . . . 5 𝑥𝐵
86, 7nfriota 7378 . . . 4 𝑥(𝑦𝐵 [𝑧 / 𝑥]𝜑)
9 sbequ12 2244 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
109riotabidv 7367 . . . 4 (𝑥 = 𝑧 → (𝑦𝐵 𝜑) = (𝑦𝐵 [𝑧 / 𝑥]𝜑))
115, 8, 10csbief 3929 . . 3 𝑧 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝑧 / 𝑥]𝜑)
124, 11vtoclg 3557 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
13 csbprc 4407 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝑦𝐵 𝜑) = ∅)
14 df-riota 7365 . . . 4 (𝑦𝐵 [𝐴 / 𝑥]𝜑) = (℩𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑))
15 euex 2572 . . . . . 6 (∃!𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑) → ∃𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑))
16 sbcex 3788 . . . . . . . 8 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
1716adantl 483 . . . . . . 7 ((𝑦𝐵[𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
1817exlimiv 1934 . . . . . 6 (∃𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
1915, 18syl 17 . . . . 5 (∃!𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
20 iotanul 6522 . . . . 5 (¬ ∃!𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑) → (℩𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑)) = ∅)
2119, 20nsyl5 159 . . . 4 𝐴 ∈ V → (℩𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑)) = ∅)
2214, 21eqtr2id 2786 . . 3 𝐴 ∈ V → ∅ = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
2313, 22eqtrd 2773 . 2 𝐴 ∈ V → 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
2412, 23pm2.61i 182 1 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397   = wceq 1542  wex 1782  [wsb 2068  wcel 2107  ∃!weu 2563  Vcvv 3475  [wsbc 3778  csb 3894  c0 4323  cio 6494  crio 7364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-in 3956  df-ss 3966  df-nul 4324  df-sn 4630  df-uni 4910  df-iota 6496  df-riota 7365
This theorem is referenced by:  cdlemkid3N  39804  cdlemkid4  39805
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