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Theorem cbviota 4899
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 1437 . . . . . 6 𝑧(𝜑𝑥 = 𝑤)
2 nfs1v 1831 . . . . . . 7 𝑥[𝑧 / 𝑥]𝜑
3 nfv 1437 . . . . . . 7 𝑥 𝑧 = 𝑤
42, 3nfbi 1497 . . . . . 6 𝑥([𝑧 / 𝑥]𝜑𝑧 = 𝑤)
5 sbequ12 1670 . . . . . . 7 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
6 equequ1 1614 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝑤𝑧 = 𝑤))
75, 6bibi12d 228 . . . . . 6 (𝑥 = 𝑧 → ((𝜑𝑥 = 𝑤) ↔ ([𝑧 / 𝑥]𝜑𝑧 = 𝑤)))
81, 4, 7cbval 1653 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑤) ↔ ∀𝑧([𝑧 / 𝑥]𝜑𝑧 = 𝑤))
9 cbviota.2 . . . . . . . 8 𝑦𝜑
109nfsb 1838 . . . . . . 7 𝑦[𝑧 / 𝑥]𝜑
11 nfv 1437 . . . . . . 7 𝑦 𝑧 = 𝑤
1210, 11nfbi 1497 . . . . . 6 𝑦([𝑧 / 𝑥]𝜑𝑧 = 𝑤)
13 nfv 1437 . . . . . 6 𝑧(𝜓𝑦 = 𝑤)
14 sbequ 1737 . . . . . . . 8 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
15 cbviota.3 . . . . . . . . 9 𝑥𝜓
16 cbviota.1 . . . . . . . . 9 (𝑥 = 𝑦 → (𝜑𝜓))
1715, 16sbie 1690 . . . . . . . 8 ([𝑦 / 𝑥]𝜑𝜓)
1814, 17syl6bb 189 . . . . . . 7 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
19 equequ1 1614 . . . . . . 7 (𝑧 = 𝑦 → (𝑧 = 𝑤𝑦 = 𝑤))
2018, 19bibi12d 228 . . . . . 6 (𝑧 = 𝑦 → (([𝑧 / 𝑥]𝜑𝑧 = 𝑤) ↔ (𝜓𝑦 = 𝑤)))
2112, 13, 20cbval 1653 . . . . 5 (∀𝑧([𝑧 / 𝑥]𝜑𝑧 = 𝑤) ↔ ∀𝑦(𝜓𝑦 = 𝑤))
228, 21bitri 177 . . . 4 (∀𝑥(𝜑𝑥 = 𝑤) ↔ ∀𝑦(𝜓𝑦 = 𝑤))
2322abbii 2169 . . 3 {𝑤 ∣ ∀𝑥(𝜑𝑥 = 𝑤)} = {𝑤 ∣ ∀𝑦(𝜓𝑦 = 𝑤)}
2423unieqi 3617 . 2 {𝑤 ∣ ∀𝑥(𝜑𝑥 = 𝑤)} = {𝑤 ∣ ∀𝑦(𝜓𝑦 = 𝑤)}
25 dfiota2 4895 . 2 (℩𝑥𝜑) = {𝑤 ∣ ∀𝑥(𝜑𝑥 = 𝑤)}
26 dfiota2 4895 . 2 (℩𝑦𝜓) = {𝑤 ∣ ∀𝑦(𝜓𝑦 = 𝑤)}
2724, 25, 263eqtr4i 2086 1 (℩𝑥𝜑) = (℩𝑦𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257   = wceq 1259  wnf 1365  [wsb 1661  {cab 2042   cuni 3607  cio 4892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-sn 3408  df-uni 3608  df-iota 4894
This theorem is referenced by:  cbviotav  4900  cbvriota  5505
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