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Theorem sb8iota 5846
Description: Variable substitution in description binder. Compare sb8eu 2501. (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 1841 . . . . . 6 𝑤(𝜑𝑥 = 𝑧)
21sb8 2422 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑤[𝑤 / 𝑥](𝜑𝑥 = 𝑧))
3 sbbi 2399 . . . . . . 7 ([𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ ([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧))
4 sb8iota.1 . . . . . . . . 9 𝑦𝜑
54nfsb 2438 . . . . . . . 8 𝑦[𝑤 / 𝑥]𝜑
6 equsb3 2430 . . . . . . . . 9 ([𝑤 / 𝑥]𝑥 = 𝑧𝑤 = 𝑧)
7 nfv 1841 . . . . . . . . 9 𝑦 𝑤 = 𝑧
86, 7nfxfr 1777 . . . . . . . 8 𝑦[𝑤 / 𝑥]𝑥 = 𝑧
95, 8nfbi 1831 . . . . . . 7 𝑦([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧)
103, 9nfxfr 1777 . . . . . 6 𝑦[𝑤 / 𝑥](𝜑𝑥 = 𝑧)
11 nfv 1841 . . . . . 6 𝑤[𝑦 / 𝑥](𝜑𝑥 = 𝑧)
12 sbequ 2374 . . . . . 6 (𝑤 = 𝑦 → ([𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ [𝑦 / 𝑥](𝜑𝑥 = 𝑧)))
1310, 11, 12cbval 2269 . . . . 5 (∀𝑤[𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ ∀𝑦[𝑦 / 𝑥](𝜑𝑥 = 𝑧))
14 equsb3 2430 . . . . . . 7 ([𝑦 / 𝑥]𝑥 = 𝑧𝑦 = 𝑧)
1514sblbis 2402 . . . . . 6 ([𝑦 / 𝑥](𝜑𝑥 = 𝑧) ↔ ([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1615albii 1745 . . . . 5 (∀𝑦[𝑦 / 𝑥](𝜑𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
172, 13, 163bitri 286 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1817abbii 2737 . . 3 {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧)}
1918unieqi 4436 . 2 {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧)}
20 dfiota2 5840 . 2 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
21 dfiota2 5840 . 2 (℩𝑦[𝑦 / 𝑥]𝜑) = {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧)}
2219, 20, 213eqtr4i 2652 1 (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1479   = wceq 1481  wnf 1706  [wsb 1878  {cab 2606   cuni 4427  cio 5837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rex 2915  df-sn 4169  df-uni 4428  df-iota 5839
This theorem is referenced by: (None)
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