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Theorem nfcrii 2928
Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
nfcri.1 𝑥𝐴
Assertion
Ref Expression
nfcrii (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem nfcrii
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfcri.1 . . . 4 𝑥𝐴
21nfcriv 2925 . . 3 𝑥 𝑧𝐴
32nf5ri 2179 . 2 (𝑧𝐴 → ∀𝑥 𝑧𝐴)
43hblem 2891 1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1599  wcel 2107  wnfc 2919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clel 2774  df-nfc 2921
This theorem is referenced by:  nfcri  2929  cleqfOLD  2964  abeq2fOLD  2967  bnj1230  31472  bnj1000  31610  bnj1204  31679  bnj1307  31690  bnj1311  31691  bnj1398  31701  bnj1466  31720  bnj1467  31721  bnj1523  31738
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