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Theorem sb8iota 4902
Description: Variable substitution in description binder. Compare sb8eu 1929. (Contributed by NM, 18-Mar-2013.)
Hypothesis
Ref Expression
sb8iota.1 𝑦𝜑
Assertion
Ref Expression
sb8iota (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb8iota
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1437 . . . . . 6 𝑤(𝜑𝑥 = 𝑧)
21sb8 1752 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑤[𝑤 / 𝑥](𝜑𝑥 = 𝑧))
3 sbbi 1849 . . . . . . 7 ([𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ ([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧))
4 sb8iota.1 . . . . . . . . 9 𝑦𝜑
54nfsb 1838 . . . . . . . 8 𝑦[𝑤 / 𝑥]𝜑
6 equsb3 1841 . . . . . . . . 9 ([𝑤 / 𝑥]𝑥 = 𝑧𝑤 = 𝑧)
7 nfv 1437 . . . . . . . . 9 𝑦 𝑤 = 𝑧
86, 7nfxfr 1379 . . . . . . . 8 𝑦[𝑤 / 𝑥]𝑥 = 𝑧
95, 8nfbi 1497 . . . . . . 7 𝑦([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧)
103, 9nfxfr 1379 . . . . . 6 𝑦[𝑤 / 𝑥](𝜑𝑥 = 𝑧)
11 nfv 1437 . . . . . 6 𝑤[𝑦 / 𝑥](𝜑𝑥 = 𝑧)
12 sbequ 1737 . . . . . 6 (𝑤 = 𝑦 → ([𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ [𝑦 / 𝑥](𝜑𝑥 = 𝑧)))
1310, 11, 12cbval 1653 . . . . 5 (∀𝑤[𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ ∀𝑦[𝑦 / 𝑥](𝜑𝑥 = 𝑧))
14 equsb3 1841 . . . . . . 7 ([𝑦 / 𝑥]𝑥 = 𝑧𝑦 = 𝑧)
1514sblbis 1850 . . . . . 6 ([𝑦 / 𝑥](𝜑𝑥 = 𝑧) ↔ ([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1615albii 1375 . . . . 5 (∀𝑦[𝑦 / 𝑥](𝜑𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
172, 13, 163bitri 199 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1817abbii 2169 . . 3 {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧)}
1918unieqi 3618 . 2 {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧)}
20 dfiota2 4896 . 2 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
21 dfiota2 4896 . 2 (℩𝑦[𝑦 / 𝑥]𝜑) = {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧)}
2219, 20, 213eqtr4i 2086 1 (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 102  wal 1257   = wceq 1259  wnf 1365  [wsb 1661  {cab 2042   cuni 3608  cio 4893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-sn 3409  df-uni 3609  df-iota 4895
This theorem is referenced by: (None)
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