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Theorem cbviotaw 6306
 Description: Change bound variables in a description binder. Version of cbviota 6308 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by Gino Giotto, 26-Jan-2024.)
Hypotheses
Ref Expression
cbviotaw.1 (𝑥 = 𝑦 → (𝜑𝜓))
cbviotaw.2 𝑦𝜑
cbviotaw.3 𝑥𝜓
Assertion
Ref Expression
cbviotaw (℩𝑥𝜑) = (℩𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbviotaw
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1915 . . . . . 6 𝑧(𝜑𝑥 = 𝑤)
2 nfs1v 2157 . . . . . . 7 𝑥[𝑧 / 𝑥]𝜑
3 nfv 1915 . . . . . . 7 𝑥 𝑧 = 𝑤
42, 3nfbi 1904 . . . . . 6 𝑥([𝑧 / 𝑥]𝜑𝑧 = 𝑤)
5 sbequ12 2250 . . . . . . 7 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
6 equequ1 2032 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝑤𝑧 = 𝑤))
75, 6bibi12d 349 . . . . . 6 (𝑥 = 𝑧 → ((𝜑𝑥 = 𝑤) ↔ ([𝑧 / 𝑥]𝜑𝑧 = 𝑤)))
81, 4, 7cbvalv1 2350 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑤) ↔ ∀𝑧([𝑧 / 𝑥]𝜑𝑧 = 𝑤))
9 cbviotaw.2 . . . . . . . 8 𝑦𝜑
109nfsbv 2338 . . . . . . 7 𝑦[𝑧 / 𝑥]𝜑
11 nfv 1915 . . . . . . 7 𝑦 𝑧 = 𝑤
1210, 11nfbi 1904 . . . . . 6 𝑦([𝑧 / 𝑥]𝜑𝑧 = 𝑤)
13 nfv 1915 . . . . . 6 𝑧(𝜓𝑦 = 𝑤)
14 cbviotaw.3 . . . . . . . 8 𝑥𝜓
15 cbviotaw.1 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
1614, 15sbhypf 3471 . . . . . . 7 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
17 equequ1 2032 . . . . . . 7 (𝑧 = 𝑦 → (𝑧 = 𝑤𝑦 = 𝑤))
1816, 17bibi12d 349 . . . . . 6 (𝑧 = 𝑦 → (([𝑧 / 𝑥]𝜑𝑧 = 𝑤) ↔ (𝜓𝑦 = 𝑤)))
1912, 13, 18cbvalv1 2350 . . . . 5 (∀𝑧([𝑧 / 𝑥]𝜑𝑧 = 𝑤) ↔ ∀𝑦(𝜓𝑦 = 𝑤))
208, 19bitri 278 . . . 4 (∀𝑥(𝜑𝑥 = 𝑤) ↔ ∀𝑦(𝜓𝑦 = 𝑤))
2120abbii 2823 . . 3 {𝑤 ∣ ∀𝑥(𝜑𝑥 = 𝑤)} = {𝑤 ∣ ∀𝑦(𝜓𝑦 = 𝑤)}
2221unieqi 4814 . 2 {𝑤 ∣ ∀𝑥(𝜑𝑥 = 𝑤)} = {𝑤 ∣ ∀𝑦(𝜓𝑦 = 𝑤)}
23 dfiota2 6300 . 2 (℩𝑥𝜑) = {𝑤 ∣ ∀𝑥(𝜑𝑥 = 𝑤)}
24 dfiota2 6300 . 2 (℩𝑦𝜓) = {𝑤 ∣ ∀𝑦(𝜓𝑦 = 𝑤)}
2522, 23, 243eqtr4i 2791 1 (℩𝑥𝜑) = (℩𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   = wceq 1538  Ⅎwnf 1785  [wsb 2069  {cab 2735  ∪ cuni 4801  ℩cio 6297 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-in 3867  df-ss 3877  df-sn 4526  df-uni 4802  df-iota 6299 This theorem is referenced by:  cbviotavw  6307  fvopab5  6796  cbvriotaw  7123
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