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Theorem cbviota 4985
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 1466 . . . . . 6 𝑧(𝜑𝑥 = 𝑤)
2 nfs1v 1863 . . . . . . 7 𝑥[𝑧 / 𝑥]𝜑
3 nfv 1466 . . . . . . 7 𝑥 𝑧 = 𝑤
42, 3nfbi 1526 . . . . . 6 𝑥([𝑧 / 𝑥]𝜑𝑧 = 𝑤)
5 sbequ12 1701 . . . . . . 7 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
6 equequ1 1645 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝑤𝑧 = 𝑤))
75, 6bibi12d 233 . . . . . 6 (𝑥 = 𝑧 → ((𝜑𝑥 = 𝑤) ↔ ([𝑧 / 𝑥]𝜑𝑧 = 𝑤)))
81, 4, 7cbval 1684 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑤) ↔ ∀𝑧([𝑧 / 𝑥]𝜑𝑧 = 𝑤))
9 cbviota.2 . . . . . . . 8 𝑦𝜑
109nfsb 1870 . . . . . . 7 𝑦[𝑧 / 𝑥]𝜑
11 nfv 1466 . . . . . . 7 𝑦 𝑧 = 𝑤
1210, 11nfbi 1526 . . . . . 6 𝑦([𝑧 / 𝑥]𝜑𝑧 = 𝑤)
13 nfv 1466 . . . . . 6 𝑧(𝜓𝑦 = 𝑤)
14 sbequ 1768 . . . . . . . 8 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
15 cbviota.3 . . . . . . . . 9 𝑥𝜓
16 cbviota.1 . . . . . . . . 9 (𝑥 = 𝑦 → (𝜑𝜓))
1715, 16sbie 1721 . . . . . . . 8 ([𝑦 / 𝑥]𝜑𝜓)
1814, 17syl6bb 194 . . . . . . 7 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
19 equequ1 1645 . . . . . . 7 (𝑧 = 𝑦 → (𝑧 = 𝑤𝑦 = 𝑤))
2018, 19bibi12d 233 . . . . . 6 (𝑧 = 𝑦 → (([𝑧 / 𝑥]𝜑𝑧 = 𝑤) ↔ (𝜓𝑦 = 𝑤)))
2112, 13, 20cbval 1684 . . . . 5 (∀𝑧([𝑧 / 𝑥]𝜑𝑧 = 𝑤) ↔ ∀𝑦(𝜓𝑦 = 𝑤))
228, 21bitri 182 . . . 4 (∀𝑥(𝜑𝑥 = 𝑤) ↔ ∀𝑦(𝜓𝑦 = 𝑤))
2322abbii 2203 . . 3 {𝑤 ∣ ∀𝑥(𝜑𝑥 = 𝑤)} = {𝑤 ∣ ∀𝑦(𝜓𝑦 = 𝑤)}
2423unieqi 3663 . 2 {𝑤 ∣ ∀𝑥(𝜑𝑥 = 𝑤)} = {𝑤 ∣ ∀𝑦(𝜓𝑦 = 𝑤)}
25 dfiota2 4981 . 2 (℩𝑥𝜑) = {𝑤 ∣ ∀𝑥(𝜑𝑥 = 𝑤)}
26 dfiota2 4981 . 2 (℩𝑦𝜓) = {𝑤 ∣ ∀𝑦(𝜓𝑦 = 𝑤)}
2724, 25, 263eqtr4i 2118 1 (℩𝑥𝜑) = (℩𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287   = wceq 1289  wnf 1394  [wsb 1692  {cab 2074   cuni 3653  cio 4978
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-sn 3452  df-uni 3654  df-iota 4980
This theorem is referenced by:  cbviotav  4986  cbvriota  5618
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