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Theorem sb8iota 6313
Description: Variable substitution in description binder. Compare sb8eu 2687. Usage of this theorem is discouraged because it depends on ax-13 2392. (Contributed by NM, 18-Mar-2013.) (New usage is discouraged.)
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 1916 . . . . . 6 𝑤(𝜑𝑥 = 𝑧)
21sb8 2561 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑤[𝑤 / 𝑥](𝜑𝑥 = 𝑧))
3 sbbi 2319 . . . . . . 7 ([𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ ([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧))
4 sb8iota.1 . . . . . . . . 9 𝑦𝜑
54nfsb 2567 . . . . . . . 8 𝑦[𝑤 / 𝑥]𝜑
6 equsb3 2110 . . . . . . . . 9 ([𝑤 / 𝑥]𝑥 = 𝑧𝑤 = 𝑧)
7 nfv 1916 . . . . . . . . 9 𝑦 𝑤 = 𝑧
86, 7nfxfr 1854 . . . . . . . 8 𝑦[𝑤 / 𝑥]𝑥 = 𝑧
95, 8nfbi 1905 . . . . . . 7 𝑦([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧)
103, 9nfxfr 1854 . . . . . 6 𝑦[𝑤 / 𝑥](𝜑𝑥 = 𝑧)
11 nfv 1916 . . . . . 6 𝑤[𝑦 / 𝑥](𝜑𝑥 = 𝑧)
12 sbequ 2091 . . . . . 6 (𝑤 = 𝑦 → ([𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ [𝑦 / 𝑥](𝜑𝑥 = 𝑧)))
1310, 11, 12cbvalv1 2363 . . . . 5 (∀𝑤[𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ ∀𝑦[𝑦 / 𝑥](𝜑𝑥 = 𝑧))
14 equsb3 2110 . . . . . . 7 ([𝑦 / 𝑥]𝑥 = 𝑧𝑦 = 𝑧)
1514sblbis 2321 . . . . . 6 ([𝑦 / 𝑥](𝜑𝑥 = 𝑧) ↔ ([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1615albii 1821 . . . . 5 (∀𝑦[𝑦 / 𝑥](𝜑𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
172, 13, 163bitri 300 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1817abbii 2889 . . 3 {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧)}
1918unieqi 4837 . 2 {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧)}
20 dfiota2 6303 . 2 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
21 dfiota2 6303 . 2 (℩𝑦[𝑦 / 𝑥]𝜑) = {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧)}
2219, 20, 213eqtr4i 2857 1 (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wal 1536   = wceq 1538  wnf 1785  [wsb 2070  {cab 2802   cuni 4824  cio 6300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-in 3926  df-ss 3936  df-sn 4550  df-uni 4825  df-iota 6302
This theorem is referenced by: (None)
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