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Theorem cbviota 4344
 Description: Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
Hypotheses
Ref Expression
cbviota.1 (x = y → (φψ))
cbviota.2 yφ
cbviota.3 xψ
Assertion
Ref Expression
cbviota (℩xφ) = (℩yψ)

Proof of Theorem cbviota
Dummy variables z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . . . . 6 z(φx = w)
2 nfs1v 2106 . . . . . . 7 x[z / x]φ
3 nfv 1619 . . . . . . 7 x z = w
42, 3nfbi 1834 . . . . . 6 x([z / x]φz = w)
5 sbequ12 1919 . . . . . . 7 (x = z → (φ ↔ [z / x]φ))
6 equequ1 1684 . . . . . . 7 (x = z → (x = wz = w))
75, 6bibi12d 312 . . . . . 6 (x = z → ((φx = w) ↔ ([z / x]φz = w)))
81, 4, 7cbval 1984 . . . . 5 (x(φx = w) ↔ z([z / x]φz = w))
9 cbviota.2 . . . . . . . 8 yφ
109nfsb 2109 . . . . . . 7 y[z / x]φ
11 nfv 1619 . . . . . . 7 y z = w
1210, 11nfbi 1834 . . . . . 6 y([z / x]φz = w)
13 nfv 1619 . . . . . 6 z(ψy = w)
14 sbequ 2060 . . . . . . . 8 (z = y → ([z / x]φ ↔ [y / x]φ))
15 cbviota.3 . . . . . . . . 9 xψ
16 cbviota.1 . . . . . . . . 9 (x = y → (φψ))
1715, 16sbie 2038 . . . . . . . 8 ([y / x]φψ)
1814, 17syl6bb 252 . . . . . . 7 (z = y → ([z / x]φψ))
19 equequ1 1684 . . . . . . 7 (z = y → (z = wy = w))
2018, 19bibi12d 312 . . . . . 6 (z = y → (([z / x]φz = w) ↔ (ψy = w)))
2112, 13, 20cbval 1984 . . . . 5 (z([z / x]φz = w) ↔ y(ψy = w))
228, 21bitri 240 . . . 4 (x(φx = w) ↔ y(ψy = w))
2322abbii 2465 . . 3 {w x(φx = w)} = {w y(ψy = w)}
2423unieqi 3901 . 2 {w x(φx = w)} = {w y(ψy = w)}
25 dfiota2 4340 . 2 (℩xφ) = {w x(φx = w)}
26 dfiota2 4340 . 2 (℩yψ) = {w y(ψy = w)}
2724, 25, 263eqtr4i 2383 1 (℩xφ) = (℩yψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ 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:  cbviotav  4345
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