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Theorem nfcrii 2755
 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 2754 . . . 4 (𝑥𝐴 → Ⅎ𝑥 𝑧𝐴)
31, 2ax-mp 5 . . 3 𝑥 𝑧𝐴
43nf5ri 2063 . 2 (𝑧𝐴 → ∀𝑥 𝑧𝐴)
54hblem 2729 1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1479  Ⅎwnf 1706   ∈ wcel 1988  Ⅎwnfc 2749 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-cleq 2613  df-clel 2616  df-nfc 2751 This theorem is referenced by:  nfcri  2756  cleqf  2787  abeq2f  2789  bnj1230  30847  bnj1000  30985  bnj1204  31054  bnj1307  31065  bnj1311  31066  bnj1398  31076  bnj1466  31095  bnj1467  31096  bnj1523  31113
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