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Theorem nfcrii 2743
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 2742 . . . 4 (𝑥𝐴 → Ⅎ𝑥 𝑧𝐴)
31, 2ax-mp 5 . . 3 𝑥 𝑧𝐴
43nf5ri 2052 . 2 (𝑧𝐴 → ∀𝑥 𝑧𝐴)
54hblem 2717 1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472  wnf 1698  wcel 1976  wnfc 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-cleq 2602  df-clel 2605  df-nfc 2739
This theorem is referenced by:  nfcri  2744  cleqf  2775  abeq2f  2777  bnj1230  29933  bnj1000  30071  bnj1204  30140  bnj1307  30151  bnj1311  30152  bnj1398  30162  bnj1466  30181  bnj1467  30182  bnj1523  30199
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