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Theorem sb8iota 4346
 Description: Variable substitution in description binder. Compare sb8eu 2222. (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 1619 . . . . . 6 w(φx = z)
21sb8 2092 . . . . 5 (x(φx = z) ↔ w[w / x](φx = z))
3 sbbi 2071 . . . . . . 7 ([w / x](φx = z) ↔ ([w / x]φ ↔ [w / x]x = z))
4 sb8iota.1 . . . . . . . . 9 yφ
54nfsb 2109 . . . . . . . 8 y[w / x]φ
6 eqsb3 2454 . . . . . . . . 9 ([w / x]x = zw = z)
7 nfv 1619 . . . . . . . . 9 y w = z
86, 7nfxfr 1570 . . . . . . . 8 y[w / x]x = z
95, 8nfbi 1834 . . . . . . 7 y([w / x]φ ↔ [w / x]x = z)
103, 9nfxfr 1570 . . . . . 6 y[w / x](φx = z)
11 nfv 1619 . . . . . 6 w[y / x](φx = z)
12 sbequ 2060 . . . . . 6 (w = y → ([w / x](φx = z) ↔ [y / x](φx = z)))
1310, 11, 12cbval 1984 . . . . 5 (w[w / x](φx = z) ↔ y[y / x](φx = z))
14 equsb3 2102 . . . . . . 7 ([y / x]x = zy = z)
1514sblbis 2072 . . . . . 6 ([y / x](φx = z) ↔ ([y / x]φy = z))
1615albii 1566 . . . . 5 (y[y / x](φx = z) ↔ y([y / x]φy = z))
172, 13, 163bitri 262 . . . 4 (x(φx = z) ↔ y([y / x]φy = z))
1817abbii 2465 . . 3 {z x(φx = z)} = {z y([y / x]φy = z)}
1918unieqi 3901 . 2 {z x(φx = z)} = {z y([y / x]φy = z)}
20 dfiota2 4340 . 2 (℩xφ) = {z x(φx = z)}
21 dfiota2 4340 . 2 (℩y[y / x]φ) = {z y([y / x]φy = z)}
2219, 20, 213eqtr4i 2383 1 (℩xφ) = (℩y[y / x]φ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642  [wsb 1648  {cab 2339  ∪cuni 3891  ℩cio 4337 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-sn 3741  df-uni 3892  df-iota 4339 This theorem is referenced by: (None)
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