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Theorem nfcrii 2906
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
2 nfcr 2905 . . . 4 (𝑥𝐴 → Ⅎ𝑥 𝑧𝐴)
31, 2ax-mp 5 . . 3 𝑥 𝑧𝐴
43nf5ri 2219 . 2 (𝑧𝐴 → ∀𝑥 𝑧𝐴)
54hblem 2880 1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629  wnf 1856  wcel 2145  wnfc 2900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clel 2767  df-nfc 2902
This theorem is referenced by:  nfcri  2907  cleqf  2939  abeq2f  2941  bnj1230  31211  bnj1000  31349  bnj1204  31418  bnj1307  31429  bnj1311  31430  bnj1398  31440  bnj1466  31459  bnj1467  31460  bnj1523  31477
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