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Theorem cbvriotaw 7103
 Description: Change bound variable in a restricted description binder. Version of cbvriota 7107 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 18-Mar-2013.) (Revised by Gino Giotto, 26-Jan-2024.)
Hypotheses
Ref Expression
cbvriotaw.1 𝑦𝜑
cbvriotaw.2 𝑥𝜓
cbvriotaw.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvriotaw (𝑥𝐴 𝜑) = (𝑦𝐴 𝜓)
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvriotaw
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eleq1w 2872 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
2 sbequ12 2250 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
31, 2anbi12d 633 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝜑) ↔ (𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)))
4 nfv 1915 . . . 4 𝑧(𝑥𝐴𝜑)
5 nfv 1915 . . . . 5 𝑥 𝑧𝐴
6 nfs1v 2157 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
75, 6nfan 1900 . . . 4 𝑥(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)
83, 4, 7cbviotaw 6291 . . 3 (℩𝑥(𝑥𝐴𝜑)) = (℩𝑧(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑))
9 eleq1w 2872 . . . . 5 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
10 cbvriotaw.2 . . . . . 6 𝑥𝜓
11 cbvriotaw.3 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
1210, 11sbhypf 3500 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
139, 12anbi12d 633 . . . 4 (𝑧 = 𝑦 → ((𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦𝐴𝜓)))
14 nfv 1915 . . . . 5 𝑦 𝑧𝐴
15 cbvriotaw.1 . . . . . 6 𝑦𝜑
1615nfsbv 2338 . . . . 5 𝑦[𝑧 / 𝑥]𝜑
1714, 16nfan 1900 . . . 4 𝑦(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)
18 nfv 1915 . . . 4 𝑧(𝑦𝐴𝜓)
1913, 17, 18cbviotaw 6291 . . 3 (℩𝑧(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)) = (℩𝑦(𝑦𝐴𝜓))
208, 19eqtri 2821 . 2 (℩𝑥(𝑥𝐴𝜑)) = (℩𝑦(𝑦𝐴𝜓))
21 df-riota 7094 . 2 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
22 df-riota 7094 . 2 (𝑦𝐴 𝜓) = (℩𝑦(𝑦𝐴𝜓))
2320, 21, 223eqtr4i 2831 1 (𝑥𝐴 𝜑) = (𝑦𝐴 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  Ⅎwnf 1785  [wsb 2069   ∈ wcel 2111  ℩cio 6282  ℩crio 7093 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 2770 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 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-sn 4526  df-uni 4802  df-iota 6284  df-riota 7094 This theorem is referenced by:  cbvriotavw  7104  disjinfi  41863
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