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Theorem nfiing 5052
Description: Bound-variable hypothesis builder for indexed intersection. Usage of this theorem is discouraged because it depends on ax-13 2374. See nfiin 5050 for a version with more disjoint variable conditions, but not requiring ax-13 2374. (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 5022 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
2 nfiung.1 . . . 4 𝑦𝐴
3 nfiung.2 . . . . 5 𝑦𝐵
43nfcri 2895 . . . 4 𝑦 𝑧𝐵
52, 4nfral 3377 . . 3 𝑦𝑥𝐴 𝑧𝐵
65nfabg 2911 . 2 𝑦{𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
71, 6nfcxfr 2902 1 𝑦 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wcel 2103  {cab 2711  wnfc 2888  wral 3063   ciin 5020
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 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-13 2374  ax-ext 2705
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 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ral 3064  df-iin 5022
This theorem is referenced by: (None)
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