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Theorem cbviotaw 6502
Description: Change bound variables in a description binder. Version of cbviota 6505 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by Andrew Salmon, 1-Aug-2011.) Avoid ax-13 2370. (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 1916 . . . . . 6 𝑧(𝜑𝑥 = 𝑤)
2 nfs1v 2152 . . . . . . 7 𝑥[𝑧 / 𝑥]𝜑
3 nfv 1916 . . . . . . 7 𝑥 𝑧 = 𝑤
42, 3nfbi 1905 . . . . . 6 𝑥([𝑧 / 𝑥]𝜑𝑧 = 𝑤)
5 sbequ12 2242 . . . . . . 7 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
6 equequ1 2027 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝑤𝑧 = 𝑤))
75, 6bibi12d 345 . . . . . 6 (𝑥 = 𝑧 → ((𝜑𝑥 = 𝑤) ↔ ([𝑧 / 𝑥]𝜑𝑧 = 𝑤)))
81, 4, 7cbvalv1 2336 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑤) ↔ ∀𝑧([𝑧 / 𝑥]𝜑𝑧 = 𝑤))
9 cbviotaw.2 . . . . . . . 8 𝑦𝜑
109nfsbv 2322 . . . . . . 7 𝑦[𝑧 / 𝑥]𝜑
11 nfv 1916 . . . . . . 7 𝑦 𝑧 = 𝑤
1210, 11nfbi 1905 . . . . . 6 𝑦([𝑧 / 𝑥]𝜑𝑧 = 𝑤)
13 nfv 1916 . . . . . 6 𝑧(𝜓𝑦 = 𝑤)
14 cbviotaw.3 . . . . . . . 8 𝑥𝜓
15 cbviotaw.1 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
1614, 15sbhypf 3538 . . . . . . 7 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
17 equequ1 2027 . . . . . . 7 (𝑧 = 𝑦 → (𝑧 = 𝑤𝑦 = 𝑤))
1816, 17bibi12d 345 . . . . . 6 (𝑧 = 𝑦 → (([𝑧 / 𝑥]𝜑𝑧 = 𝑤) ↔ (𝜓𝑦 = 𝑤)))
1912, 13, 18cbvalv1 2336 . . . . 5 (∀𝑧([𝑧 / 𝑥]𝜑𝑧 = 𝑤) ↔ ∀𝑦(𝜓𝑦 = 𝑤))
208, 19bitri 275 . . . 4 (∀𝑥(𝜑𝑥 = 𝑤) ↔ ∀𝑦(𝜓𝑦 = 𝑤))
2120abbii 2801 . . 3 {𝑤 ∣ ∀𝑥(𝜑𝑥 = 𝑤)} = {𝑤 ∣ ∀𝑦(𝜓𝑦 = 𝑤)}
2221unieqi 4921 . 2 {𝑤 ∣ ∀𝑥(𝜑𝑥 = 𝑤)} = {𝑤 ∣ ∀𝑦(𝜓𝑦 = 𝑤)}
23 dfiota2 6496 . 2 (℩𝑥𝜑) = {𝑤 ∣ ∀𝑥(𝜑𝑥 = 𝑤)}
24 dfiota2 6496 . 2 (℩𝑦𝜓) = {𝑤 ∣ ∀𝑦(𝜓𝑦 = 𝑤)}
2522, 23, 243eqtr4i 2769 1 (℩𝑥𝜑) = (℩𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1538   = wceq 1540  wnf 1784  [wsb 2066  {cab 2708   cuni 4908  cio 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3955  df-ss 3965  df-sn 4629  df-uni 4909  df-iota 6495
This theorem is referenced by:  cbviotavwOLD  6504  fvopab5  7030  cbvriotaw  7377
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