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Theorem cbvriota 5418
 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 (x A φ) = (y A 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 2097 . . . . 5 (x = z → (x Az A))
2 sbequ12 1651 . . . . 5 (x = z → (φ ↔ [z / x]φ))
31, 2anbi12d 442 . . . 4 (x = z → ((x A φ) ↔ (z A [z / x]φ)))
4 nfv 1418 . . . 4 z(x A φ)
5 nfv 1418 . . . . 5 x z A
6 nfs1v 1812 . . . . 5 x[z / x]φ
75, 6nfan 1454 . . . 4 x(z A [z / x]φ)
83, 4, 7cbviota 4814 . . 3 (℩x(x A φ)) = (℩z(z A [z / x]φ))
9 eleq1 2097 . . . . 5 (z = y → (z Ay A))
10 sbequ 1718 . . . . . 6 (z = y → ([z / x]φ ↔ [y / x]φ))
11 cbvriota.2 . . . . . . 7 xψ
12 cbvriota.3 . . . . . . 7 (x = y → (φψ))
1311, 12sbie 1671 . . . . . 6 ([y / x]φψ)
1410, 13syl6bb 185 . . . . 5 (z = y → ([z / x]φψ))
159, 14anbi12d 442 . . . 4 (z = y → ((z A [z / x]φ) ↔ (y A ψ)))
16 nfv 1418 . . . . 5 y z A
17 cbvriota.1 . . . . . 6 yφ
1817nfsb 1819 . . . . 5 y[z / x]φ
1916, 18nfan 1454 . . . 4 y(z A [z / x]φ)
20 nfv 1418 . . . 4 z(y A ψ)
2115, 19, 20cbviota 4814 . . 3 (℩z(z A [z / x]φ)) = (℩y(y A ψ))
228, 21eqtri 2057 . 2 (℩x(x A φ)) = (℩y(y A ψ))
23 df-riota 5409 . 2 (x A φ) = (℩x(x A φ))
24 df-riota 5409 . 2 (y A ψ) = (℩y(y A ψ))
2522, 23, 243eqtr4i 2067 1 (x A φ) = (y A ψ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  Ⅎwnf 1346   ∈ wcel 1390  [wsb 1642  ℩cio 4807  ℩crio 5408 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-sn 3372  df-uni 3571  df-iota 4809  df-riota 5409 This theorem is referenced by:  cbvriotav  5419
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