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Theorem nfcrii 2786
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 2785 . . . 4 (𝑥𝐴 → Ⅎ𝑥 𝑧𝐴)
31, 2ax-mp 5 . . 3 𝑥 𝑧𝐴
43nf5ri 2103 . 2 (𝑧𝐴 → ∀𝑥 𝑧𝐴)
54hblem 2760 1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1521  wnf 1748  wcel 2030  wnfc 2780
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-cleq 2644  df-clel 2647  df-nfc 2782
This theorem is referenced by:  nfcri  2787  cleqf  2819  abeq2f  2821  bnj1230  30999  bnj1000  31137  bnj1204  31206  bnj1307  31217  bnj1311  31218  bnj1398  31228  bnj1466  31247  bnj1467  31248  bnj1523  31265
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