Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbviing Structured version   Visualization version   GIF version

Theorem cbviing 4927
 Description: Change bound variables in an indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2379. See cbviin 4925 for a version with more disjoint variable conditions, but not requiring ax-13 2379. (Contributed by Jeff Hankins, 26-Aug-2009.) (Revised by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbviung.1 𝑦𝐵
cbviung.2 𝑥𝐶
cbviung.3 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbviing 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Distinct variable groups:   𝑦,𝐴   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem cbviing
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 cbviung.1 . . . . 5 𝑦𝐵
21nfcri 2943 . . . 4 𝑦 𝑧𝐵
3 cbviung.2 . . . . 5 𝑥𝐶
43nfcri 2943 . . . 4 𝑥 𝑧𝐶
5 cbviung.3 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
65eleq2d 2875 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
72, 4, 6cbvral 3392 . . 3 (∀𝑥𝐴 𝑧𝐵 ↔ ∀𝑦𝐴 𝑧𝐶)
87abbii 2863 . 2 {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵} = {𝑧 ∣ ∀𝑦𝐴 𝑧𝐶}
9 df-iin 4885 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
10 df-iin 4885 . 2 𝑦𝐴 𝐶 = {𝑧 ∣ ∀𝑦𝐴 𝑧𝐶}
118, 9, 103eqtr4i 2831 1 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {cab 2776  Ⅎwnfc 2936  ∀wral 3106  ∩ ciin 4883 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-13 2379  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-nfc 2938  df-ral 3111  df-iin 4885 This theorem is referenced by:  cbviinvg  4931
 Copyright terms: Public domain W3C validator