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Theorem cbvriota 6817
Description: Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
cbvriota.1 𝑦𝜑
cbvriota.2 𝑥𝜓
cbvriota.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvriota (𝑥𝐴 𝜑) = (𝑦𝐴 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvriota
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eleq1w 2827 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
2 sbequ12 2278 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
31, 2anbi12d 624 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝜑) ↔ (𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)))
4 nfv 2009 . . . 4 𝑧(𝑥𝐴𝜑)
5 nfv 2009 . . . . 5 𝑥 𝑧𝐴
6 nfs1v 2287 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
75, 6nfan 1998 . . . 4 𝑥(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)
83, 4, 7cbviota 6038 . . 3 (℩𝑥(𝑥𝐴𝜑)) = (℩𝑧(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑))
9 eleq1w 2827 . . . . 5 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
10 sbequ 2467 . . . . . 6 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
11 cbvriota.2 . . . . . . 7 𝑥𝜓
12 cbvriota.3 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
1311, 12sbie 2499 . . . . . 6 ([𝑦 / 𝑥]𝜑𝜓)
1410, 13syl6bb 278 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
159, 14anbi12d 624 . . . 4 (𝑧 = 𝑦 → ((𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦𝐴𝜓)))
16 nfv 2009 . . . . 5 𝑦 𝑧𝐴
17 cbvriota.1 . . . . . 6 𝑦𝜑
1817nfsb 2534 . . . . 5 𝑦[𝑧 / 𝑥]𝜑
1916, 18nfan 1998 . . . 4 𝑦(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)
20 nfv 2009 . . . 4 𝑧(𝑦𝐴𝜓)
2115, 19, 20cbviota 6038 . . 3 (℩𝑧(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)) = (℩𝑦(𝑦𝐴𝜓))
228, 21eqtri 2787 . 2 (℩𝑥(𝑥𝐴𝜑)) = (℩𝑦(𝑦𝐴𝜓))
23 df-riota 6807 . 2 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
24 df-riota 6807 . 2 (𝑦𝐴 𝜓) = (℩𝑦(𝑦𝐴𝜓))
2522, 23, 243eqtr4i 2797 1 (𝑥𝐴 𝜑) = (𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wnf 1878  [wsb 2062  wcel 2155  cio 6031  crio 6806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rex 3061  df-sn 4337  df-uni 4597  df-iota 6033  df-riota 6807
This theorem is referenced by:  cbvriotav  6818  disjinfi  40053
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