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Theorem nfiin 4701
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 4675 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
2 nfiun.1 . . . 4 𝑦𝐴
3 nfiun.2 . . . . 5 𝑦𝐵
43nfcri 2896 . . . 4 𝑦 𝑧𝐵
52, 4nfral 3083 . . 3 𝑦𝑥𝐴 𝑧𝐵
65nfab 2907 . 2 𝑦{𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
71, 6nfcxfr 2900 1 𝑦 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wcel 2139  {cab 2746  wnfc 2889  wral 3050   ciin 4673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-iin 4675
This theorem is referenced by:  iinab  4733  fnlimcnv  40420  fnlimfvre  40427  fnlimabslt  40432  iinhoiicc  41412  preimageiingt  41454  preimaleiinlt  41455  smflimlem6  41508  smflim  41509  smflim2  41536  smfsup  41544  smfsupmpt  41545  smfsupxr  41546  smfinflem  41547  smfinf  41548  smfinfmpt  41549  smflimsup  41558  smfliminf  41561
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