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

Theorem cbviing 4948
Description: Change bound variables in an indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2371. See cbviin 4946 for a version with more disjoint variable conditions, but not requiring ax-13 2371. (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 2891 . . . 4 𝑦 𝑧𝐵
3 cbviung.2 . . . . 5 𝑥𝐶
43nfcri 2891 . . . 4 𝑥 𝑧𝐶
5 cbviung.3 . . . . 5 (𝑥 = 𝑦𝐵 = 𝐶)
65eleq2d 2823 . . . 4 (𝑥 = 𝑦 → (𝑧𝐵𝑧𝐶))
72, 4, 6cbvral 3354 . . 3 (∀𝑥𝐴 𝑧𝐵 ↔ ∀𝑦𝐴 𝑧𝐶)
87abbii 2808 . 2 {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵} = {𝑧 ∣ ∀𝑦𝐴 𝑧𝐶}
9 df-iin 4907 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
10 df-iin 4907 . 2 𝑦𝐴 𝐶 = {𝑧 ∣ ∀𝑦𝐴 𝑧𝐶}
118, 9, 103eqtr4i 2775 1 𝑥𝐴 𝐵 = 𝑦𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  {cab 2714  wnfc 2884  wral 3061   ciin 4905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-13 2371  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-iin 4907
This theorem is referenced by:  cbviinvg  4952
  Copyright terms: Public domain W3C validator