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Theorem sb8iota 4817
Description: Variable substitution in description binder. Compare sb8eu 1910. (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 1418 . . . . . 6 w(φx = z)
21sb8 1733 . . . . 5 (x(φx = z) ↔ w[w / x](φx = z))
3 sbbi 1830 . . . . . . 7 ([w / x](φx = z) ↔ ([w / x]φ ↔ [w / x]x = z))
4 sb8iota.1 . . . . . . . . 9 yφ
54nfsb 1819 . . . . . . . 8 y[w / x]φ
6 equsb3 1822 . . . . . . . . 9 ([w / x]x = zw = z)
7 nfv 1418 . . . . . . . . 9 y w = z
86, 7nfxfr 1360 . . . . . . . 8 y[w / x]x = z
95, 8nfbi 1478 . . . . . . 7 y([w / x]φ ↔ [w / x]x = z)
103, 9nfxfr 1360 . . . . . 6 y[w / x](φx = z)
11 nfv 1418 . . . . . 6 w[y / x](φx = z)
12 sbequ 1718 . . . . . 6 (w = y → ([w / x](φx = z) ↔ [y / x](φx = z)))
1310, 11, 12cbval 1634 . . . . 5 (w[w / x](φx = z) ↔ y[y / x](φx = z))
14 equsb3 1822 . . . . . . 7 ([y / x]x = zy = z)
1514sblbis 1831 . . . . . 6 ([y / x](φx = z) ↔ ([y / x]φy = z))
1615albii 1356 . . . . 5 (y[y / x](φx = z) ↔ y([y / x]φy = z))
172, 13, 163bitri 195 . . . 4 (x(φx = z) ↔ y([y / x]φy = z))
1817abbii 2150 . . 3 {zx(φx = z)} = {zy([y / x]φy = z)}
1918unieqi 3581 . 2 {zx(φx = z)} = {zy([y / x]φy = z)}
20 dfiota2 4811 . 2 (℩xφ) = {zx(φx = z)}
21 dfiota2 4811 . 2 (℩y[y / x]φ) = {zy([y / x]φy = z)}
2219, 20, 213eqtr4i 2067 1 (℩xφ) = (℩y[y / x]φ)
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1240   = wceq 1242  wnf 1346  [wsb 1642  {cab 2023   cuni 3571  cio 4808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-sn 3373  df-uni 3572  df-iota 4810
This theorem is referenced by: (None)
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