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Theorem csbriota 6577
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 3517 . . . 4 (𝑧 = 𝐴𝑧 / 𝑥(𝑦𝐵 𝜑) = 𝐴 / 𝑥(𝑦𝐵 𝜑))
2 dfsbcq2 3420 . . . . 5 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32riotabidv 6567 . . . 4 (𝑧 = 𝐴 → (𝑦𝐵 [𝑧 / 𝑥]𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
41, 3eqeq12d 2636 . . 3 (𝑧 = 𝐴 → (𝑧 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝑧 / 𝑥]𝜑) ↔ 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑)))
5 vex 3189 . . . 4 𝑧 ∈ V
6 nfs1v 2436 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
7 nfcv 2761 . . . . 5 𝑥𝐵
86, 7nfriota 6574 . . . 4 𝑥(𝑦𝐵 [𝑧 / 𝑥]𝜑)
9 sbequ12 2108 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
109riotabidv 6567 . . . 4 (𝑥 = 𝑧 → (𝑦𝐵 𝜑) = (𝑦𝐵 [𝑧 / 𝑥]𝜑))
115, 8, 10csbief 3539 . . 3 𝑧 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝑧 / 𝑥]𝜑)
124, 11vtoclg 3252 . 2 (𝐴 ∈ V → 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
13 csbprc 3952 . . 3 𝐴 ∈ V → 𝐴 / 𝑥(𝑦𝐵 𝜑) = ∅)
14 df-riota 6565 . . . 4 (𝑦𝐵 [𝐴 / 𝑥]𝜑) = (℩𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑))
15 euex 2493 . . . . . . 7 (∃!𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑) → ∃𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑))
16 sbcex 3427 . . . . . . . . 9 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
1716adantl 482 . . . . . . . 8 ((𝑦𝐵[𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
1817exlimiv 1855 . . . . . . 7 (∃𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
1915, 18syl 17 . . . . . 6 (∃!𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑) → 𝐴 ∈ V)
2019con3i 150 . . . . 5 𝐴 ∈ V → ¬ ∃!𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑))
21 iotanul 5825 . . . . 5 (¬ ∃!𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑) → (℩𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑)) = ∅)
2220, 21syl 17 . . . 4 𝐴 ∈ V → (℩𝑦(𝑦𝐵[𝐴 / 𝑥]𝜑)) = ∅)
2314, 22syl5req 2668 . . 3 𝐴 ∈ V → ∅ = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
2413, 23eqtrd 2655 . 2 𝐴 ∈ V → 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
2512, 24pm2.61i 176 1 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1480  wex 1701  [wsb 1877  wcel 1987  ∃!weu 2469  Vcvv 3186  [wsbc 3417  csb 3514  c0 3891  cio 5808  crio 6564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-in 3562  df-ss 3569  df-nul 3892  df-sn 4149  df-uni 4403  df-iota 5810  df-riota 6565
This theorem is referenced by:  cdlemkid3N  35698  cdlemkid4  35699
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