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Theorem nfcrii 2967
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 . . . . 5 𝑥𝐴
21nfcriv 2964 . . . 4 𝑥 𝑧𝐴
32nfsbv 2350 . . 3 𝑥[𝑦 / 𝑧]𝑧𝐴
43nf5ri 2196 . 2 ([𝑦 / 𝑧]𝑧𝐴 → ∀𝑥[𝑦 / 𝑧]𝑧𝐴)
5 clelsb3 2939 . 2 ([𝑦 / 𝑧]𝑧𝐴𝑦𝐴)
65albii 1821 . 2 (∀𝑥[𝑦 / 𝑧]𝑧𝐴 ↔ ∀𝑥 𝑦𝐴)
74, 5, 63imtr3i 294 1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  [wsb 2070  wcel 2115  wnfc 2958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-10 2146  ax-11 2162  ax-12 2178
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2071  df-clel 2892  df-nfc 2960
This theorem is referenced by:  nfcri  2968  bnj1230  32081  bnj1000  32220  bnj1204  32291  bnj1307  32302  bnj1311  32303  bnj1398  32313  bnj1466  32332  bnj1467  32333  bnj1523  32350
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