MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfii1 Structured version   Visualization version   GIF version

Theorem nfii1 4583
Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.)
Assertion
Ref Expression
nfii1 𝑥 𝑥𝐴 𝐵

Proof of Theorem nfii1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-iin 4555 . 2 𝑥𝐴 𝐵 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}
2 nfra1 2970 . . 3 𝑥𝑥𝐴 𝑦𝐵
32nfab 2798 . 2 𝑥{𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}
41, 3nfcxfr 2791 1 𝑥 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wcel 2030  {cab 2637  wnfc 2780  wral 2941   ciin 4553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-iin 4555
This theorem is referenced by:  dmiin  5401  scott0  8787  gruiin  9670  iinssiin  39626  iooiinicc  40087  iooiinioc  40101  fnlimfvre  40224  fnlimabslt  40229  meaiininclem  41021  hspdifhsp  41151  smflimlem2  41301  smflim  41306  smflimmpt  41337  smfsuplem1  41338  smfsupmpt  41342  smfsupxr  41343  smfinflem  41344  smfinfmpt  41346  smflimsuplem7  41353  smflimsuplem8  41354  smflimsupmpt  41356  smfliminfmpt  41359
  Copyright terms: Public domain W3C validator