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Theorem nfiin 4520
Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.)
Hypotheses
Ref Expression
nfiun.1 𝑦𝐴
nfiun.2 𝑦𝐵
Assertion
Ref Expression
nfiin 𝑦 𝑥𝐴 𝐵

Proof of Theorem nfiin
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-iin 4493 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
2 nfiun.1 . . . 4 𝑦𝐴
3 nfiun.2 . . . . 5 𝑦𝐵
43nfcri 2755 . . . 4 𝑦 𝑧𝐵
52, 4nfral 2940 . . 3 𝑦𝑥𝐴 𝑧𝐵
65nfab 2765 . 2 𝑦{𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
71, 6nfcxfr 2759 1 𝑦 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wcel 1987  {cab 2607  wnfc 2748  wral 2907   ciin 4491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-iin 4493
This theorem is referenced by:  iinab  4552  fnlimcnv  39331  fnlimfvre  39338  fnlimabslt  39343  iinhoiicc  40221  preimageiingt  40263  preimaleiinlt  40264  smflimlem6  40317  smflim  40318  smflim2  40345  smfsup  40353  smfsupmpt  40354  smfsupxr  40355  smfinflem  40356  smfinf  40357  smfinfmpt  40358  smflimsup  40367
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