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Theorem bj-nfcf 32895
Description: Version of df-nfc 2751 with a dv condition replaced with a non-freeness hypothesis. (Contributed by BJ, 2-May-2019.)
Hypothesis
Ref Expression
bj-nfcf.nf 𝑦𝐴
Assertion
Ref Expression
bj-nfcf (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem bj-nfcf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-nfc 2751 . 2 (𝑥𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)
2 bj-nfcf.nf . . . . . 6 𝑦𝐴
32nfcri 2756 . . . . 5 𝑦 𝑧𝐴
43nfnf 2156 . . . 4 𝑦𝑥 𝑧𝐴
54sb8 2422 . . 3 (∀𝑧𝑥 𝑧𝐴 ↔ ∀𝑦[𝑦 / 𝑧]Ⅎ𝑥 𝑧𝐴)
6 bj-sbnf 32803 . . . . 5 ([𝑦 / 𝑧]Ⅎ𝑥 𝑧𝐴 ↔ Ⅎ𝑥[𝑦 / 𝑧]𝑧𝐴)
7 clelsb3 2727 . . . . . 6 ([𝑦 / 𝑧]𝑧𝐴𝑦𝐴)
87nfbii 1776 . . . . 5 (Ⅎ𝑥[𝑦 / 𝑧]𝑧𝐴 ↔ Ⅎ𝑥 𝑦𝐴)
96, 8bitri 264 . . . 4 ([𝑦 / 𝑧]Ⅎ𝑥 𝑧𝐴 ↔ Ⅎ𝑥 𝑦𝐴)
109albii 1745 . . 3 (∀𝑦[𝑦 / 𝑧]Ⅎ𝑥 𝑧𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
115, 10bitri 264 . 2 (∀𝑧𝑥 𝑧𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
121, 11bitri 264 1 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wal 1479  wnf 1706  [wsb 1878  wcel 1988  wnfc 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: (None)
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