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Theorem cbvriota 6586
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 eleq1 2686 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
2 sbequ12 2108 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
31, 2anbi12d 746 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝜑) ↔ (𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)))
4 nfv 1840 . . . 4 𝑧(𝑥𝐴𝜑)
5 nfv 1840 . . . . 5 𝑥 𝑧𝐴
6 nfs1v 2436 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
75, 6nfan 1825 . . . 4 𝑥(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)
83, 4, 7cbviota 5825 . . 3 (℩𝑥(𝑥𝐴𝜑)) = (℩𝑧(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑))
9 eleq1 2686 . . . . 5 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
10 sbequ 2375 . . . . . 6 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
11 cbvriota.2 . . . . . . 7 𝑥𝜓
12 cbvriota.3 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
1311, 12sbie 2407 . . . . . 6 ([𝑦 / 𝑥]𝜑𝜓)
1410, 13syl6bb 276 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
159, 14anbi12d 746 . . . 4 (𝑧 = 𝑦 → ((𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦𝐴𝜓)))
16 nfv 1840 . . . . 5 𝑦 𝑧𝐴
17 cbvriota.1 . . . . . 6 𝑦𝜑
1817nfsb 2439 . . . . 5 𝑦[𝑧 / 𝑥]𝜑
1916, 18nfan 1825 . . . 4 𝑦(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)
20 nfv 1840 . . . 4 𝑧(𝑦𝐴𝜓)
2115, 19, 20cbviota 5825 . . 3 (℩𝑧(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)) = (℩𝑦(𝑦𝐴𝜓))
228, 21eqtri 2643 . 2 (℩𝑥(𝑥𝐴𝜑)) = (℩𝑦(𝑦𝐴𝜓))
23 df-riota 6576 . 2 (𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
24 df-riota 6576 . 2 (𝑦𝐴 𝜓) = (℩𝑦(𝑦𝐴𝜓))
2522, 23, 243eqtr4i 2653 1 (𝑥𝐴 𝜑) = (𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wnf 1705  [wsb 1877  wcel 1987  cio 5818  crio 6575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2914  df-sn 4156  df-uni 4410  df-iota 5820  df-riota 6576
This theorem is referenced by:  cbvriotav  6587  disjinfi  38889
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