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Theorem csbriotag 5441
Description: Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)
Assertion
Ref Expression
csbriotag (𝐴𝑉𝐴 / 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)   𝑉(𝑥,𝑦)

Proof of Theorem csbriotag
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 2852 . . 3 (𝑧 = 𝐴𝑧 / 𝑥(𝑦𝐵 𝜑) = 𝐴 / 𝑥(𝑦𝐵 𝜑))
2 dfsbcq2 2764 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32riotabidv 5431 . . 3 (𝑧 = 𝐴 → (𝑦𝐵 [𝑧 / 𝑥]𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
41, 3eqeq12d 2054 . 2 (𝑧 = 𝐴 → (𝑧 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝑧 / 𝑥]𝜑) ↔ 𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑)))
5 vex 2557 . . 3 𝑧 ∈ V
6 nfs1v 1815 . . . 4 𝑥[𝑧 / 𝑥]𝜑
7 nfcv 2178 . . . 4 𝑥𝐵
86, 7nfriota 5438 . . 3 𝑥(𝑦𝐵 [𝑧 / 𝑥]𝜑)
9 sbequ12 1654 . . . 4 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
109riotabidv 5431 . . 3 (𝑥 = 𝑧 → (𝑦𝐵 𝜑) = (𝑦𝐵 [𝑧 / 𝑥]𝜑))
115, 8, 10csbief 2888 . 2 𝑧 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝑧 / 𝑥]𝜑)
124, 11vtoclg 2610 1 (𝐴𝑉𝐴 / 𝑥(𝑦𝐵 𝜑) = (𝑦𝐵 [𝐴 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  wcel 1393  [wsb 1645  [wsbc 2761  csb 2849  crio 5428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2309  df-v 2556  df-sbc 2762  df-csb 2850  df-sn 3378  df-uni 3577  df-iota 4828  df-riota 5429
This theorem is referenced by: (None)
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