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Theorem nfcrii 2639
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 2638 . . . 4 (𝑥𝐴 → Ⅎ𝑥 𝑧𝐴)
31, 2ax-mp 5 . . 3 𝑥 𝑧𝐴
43nfri 2005 . 2 (𝑧𝐴 → ∀𝑥 𝑧𝐴)
54hblem 2613 1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1466  wnf 1696  wcel 1937  wnfc 2633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-ex 1693  df-nf 1697  df-sb 1829  df-cleq 2498  df-clel 2501  df-nfc 2635
This theorem is referenced by:  nfcri  2640  cleqf  2671  abeq2f  2673  bnj1230  29766  bnj1000  29904  bnj1204  29973  bnj1307  29984  bnj1311  29985  bnj1398  29995  bnj1466  30014  bnj1467  30015  bnj1523  30032
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