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Theorem cbviin 3695
 Description: Change bound variables in an indexed intersection. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
cbviun.1 𝑦𝐵
cbviun.2 𝑥𝐶
cbviun.3 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbviin 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Distinct variable groups:   𝑦,𝐴   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem cbviin
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbviun.1 . . . . 5 𝑦𝐵
21nfcri 2172 . . . 4 𝑦 𝑧𝐵
3 cbviun.2 . . . . 5 𝑥𝐶
43nfcri 2172 . . . 4 𝑥 𝑧𝐶
5 cbviun.3 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
65eleq2d 2107 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
72, 4, 6cbvral 2529 . . 3 (∀𝑥𝐴 𝑧𝐵 ↔ ∀𝑦𝐴 𝑧𝐶)
87abbii 2153 . 2 {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵} = {𝑧 ∣ ∀𝑦𝐴 𝑧𝐶}
9 df-iin 3660 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
10 df-iin 3660 . 2 𝑦𝐴 𝐶 = {𝑧 ∣ ∀𝑦𝐴 𝑧𝐶}
118, 9, 103eqtr4i 2070 1 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  {cab 2026  Ⅎwnfc 2165  ∀wral 2306  ∩ ciin 3658 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-iin 3660 This theorem is referenced by:  cbviinv  3697
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