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Theorem nfiing 5049
Description: Bound-variable hypothesis builder for indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2380. See nfiin 5047 for a version with more disjoint variable conditions, but not requiring ax-13 2380. (Contributed by Mario Carneiro, 25-Jan-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfiung.1 𝑦𝐴
nfiung.2 𝑦𝐵
Assertion
Ref Expression
nfiing 𝑦 𝑥𝐴 𝐵

Proof of Theorem nfiing
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-iin 5018 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
2 nfiung.1 . . . 4 𝑦𝐴
3 nfiung.2 . . . . 5 𝑦𝐵
43nfcri 2900 . . . 4 𝑦 𝑧𝐵
52, 4nfral 3382 . . 3 𝑦𝑥𝐴 𝑧𝐵
65nfabg 2915 . 2 𝑦{𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
71, 6nfcxfr 2906 1 𝑦 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wcel 2108  {cab 2717  wnfc 2893  wral 3067   ciin 5016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-13 2380  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-iin 5018
This theorem is referenced by: (None)
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