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Theorem sb8iota 5053
Description: Variable substitution in description binder. Compare sb8eu 1988. (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 1491 . . . . . 6 𝑤(𝜑𝑥 = 𝑧)
21sb8 1810 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑤[𝑤 / 𝑥](𝜑𝑥 = 𝑧))
3 sbbi 1908 . . . . . . 7 ([𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ ([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧))
4 sb8iota.1 . . . . . . . . 9 𝑦𝜑
54nfsb 1897 . . . . . . . 8 𝑦[𝑤 / 𝑥]𝜑
6 equsb3 1900 . . . . . . . . 9 ([𝑤 / 𝑥]𝑥 = 𝑧𝑤 = 𝑧)
7 nfv 1491 . . . . . . . . 9 𝑦 𝑤 = 𝑧
86, 7nfxfr 1433 . . . . . . . 8 𝑦[𝑤 / 𝑥]𝑥 = 𝑧
95, 8nfbi 1551 . . . . . . 7 𝑦([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧)
103, 9nfxfr 1433 . . . . . 6 𝑦[𝑤 / 𝑥](𝜑𝑥 = 𝑧)
11 nfv 1491 . . . . . 6 𝑤[𝑦 / 𝑥](𝜑𝑥 = 𝑧)
12 sbequ 1794 . . . . . 6 (𝑤 = 𝑦 → ([𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ [𝑦 / 𝑥](𝜑𝑥 = 𝑧)))
1310, 11, 12cbval 1710 . . . . 5 (∀𝑤[𝑤 / 𝑥](𝜑𝑥 = 𝑧) ↔ ∀𝑦[𝑦 / 𝑥](𝜑𝑥 = 𝑧))
14 equsb3 1900 . . . . . . 7 ([𝑦 / 𝑥]𝑥 = 𝑧𝑦 = 𝑧)
1514sblbis 1909 . . . . . 6 ([𝑦 / 𝑥](𝜑𝑥 = 𝑧) ↔ ([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1615albii 1429 . . . . 5 (∀𝑦[𝑦 / 𝑥](𝜑𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
172, 13, 163bitri 205 . . . 4 (∀𝑥(𝜑𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧))
1817abbii 2230 . . 3 {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧)}
1918unieqi 3712 . 2 {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)} = {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧)}
20 dfiota2 5047 . 2 (℩𝑥𝜑) = {𝑧 ∣ ∀𝑥(𝜑𝑥 = 𝑧)}
21 dfiota2 5047 . 2 (℩𝑦[𝑦 / 𝑥]𝜑) = {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑𝑦 = 𝑧)}
2219, 20, 213eqtr4i 2145 1 (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 104  wal 1312   = wceq 1314  wnf 1419  [wsb 1718  {cab 2101   cuni 3702  cio 5044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rex 2396  df-sn 3499  df-uni 3703  df-iota 5046
This theorem is referenced by: (None)
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