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Theorem csbiotag 5116
 Description: Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.)
Assertion
Ref Expression
csbiotag (𝐴𝑉𝐴 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem csbiotag
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3006 . . 3 (𝑧 = 𝐴𝑧 / 𝑥(℩𝑦𝜑) = 𝐴 / 𝑥(℩𝑦𝜑))
2 dfsbcq2 2912 . . . 4 (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32iotabidv 5109 . . 3 (𝑧 = 𝐴 → (℩𝑦[𝑧 / 𝑥]𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
41, 3eqeq12d 2154 . 2 (𝑧 = 𝐴 → (𝑧 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑) ↔ 𝐴 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)))
5 vex 2689 . . 3 𝑧 ∈ V
6 nfs1v 1912 . . . 4 𝑥[𝑧 / 𝑥]𝜑
76nfiotaw 5092 . . 3 𝑥(℩𝑦[𝑧 / 𝑥]𝜑)
8 sbequ12 1744 . . . 4 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
98iotabidv 5109 . . 3 (𝑥 = 𝑧 → (℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑))
105, 7, 9csbief 3044 . 2 𝑧 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑)
114, 10vtoclg 2746 1 (𝐴𝑉𝐴 / 𝑥(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331   ∈ wcel 1480  [wsb 1735  [wsbc 2909  ⦋csb 3003  ℩cio 5086 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-sn 3533  df-uni 3737  df-iota 5088 This theorem is referenced by:  csbfv12g  5457
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