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Theorem cbviota 4815
 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 1418 . . . . . 6 z(φx = w)
2 nfs1v 1812 . . . . . . 7 x[z / x]φ
3 nfv 1418 . . . . . . 7 x z = w
42, 3nfbi 1478 . . . . . 6 x([z / x]φz = w)
5 sbequ12 1651 . . . . . . 7 (x = z → (φ ↔ [z / x]φ))
6 equequ1 1595 . . . . . . 7 (x = z → (x = wz = w))
75, 6bibi12d 224 . . . . . 6 (x = z → ((φx = w) ↔ ([z / x]φz = w)))
81, 4, 7cbval 1634 . . . . 5 (x(φx = w) ↔ z([z / x]φz = w))
9 cbviota.2 . . . . . . . 8 yφ
109nfsb 1819 . . . . . . 7 y[z / x]φ
11 nfv 1418 . . . . . . 7 y z = w
1210, 11nfbi 1478 . . . . . 6 y([z / x]φz = w)
13 nfv 1418 . . . . . 6 z(ψy = w)
14 sbequ 1718 . . . . . . . 8 (z = y → ([z / x]φ ↔ [y / x]φ))
15 cbviota.3 . . . . . . . . 9 xψ
16 cbviota.1 . . . . . . . . 9 (x = y → (φψ))
1715, 16sbie 1671 . . . . . . . 8 ([y / x]φψ)
1814, 17syl6bb 185 . . . . . . 7 (z = y → ([z / x]φψ))
19 equequ1 1595 . . . . . . 7 (z = y → (z = wy = w))
2018, 19bibi12d 224 . . . . . 6 (z = y → (([z / x]φz = w) ↔ (ψy = w)))
2112, 13, 20cbval 1634 . . . . 5 (z([z / x]φz = w) ↔ y(ψy = w))
228, 21bitri 173 . . . 4 (x(φx = w) ↔ y(ψy = w))
2322abbii 2150 . . 3 {wx(φx = w)} = {wy(ψy = w)}
2423unieqi 3581 . 2 {wx(φx = w)} = {wy(ψy = w)}
25 dfiota2 4811 . 2 (℩xφ) = {wx(φx = w)}
26 dfiota2 4811 . 2 (℩yψ) = {wy(ψy = w)}
2724, 25, 263eqtr4i 2067 1 (℩xφ) = (℩yψ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240   = wceq 1242  Ⅎwnf 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:  cbviotav  4816  cbvriota  5421
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