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Theorem bj-nfcf 32112
 Description: Version of df-nfc 2740 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 2740 . 2 (𝑥𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)
2 bj-nfcf.nf . . . . . 6 𝑦𝐴
32nfcri 2745 . . . . 5 𝑦 𝑧𝐴
43nfnf 2144 . . . 4 𝑦𝑥 𝑧𝐴
54sb8 2412 . . 3 (∀𝑧𝑥 𝑧𝐴 ↔ ∀𝑦[𝑦 / 𝑧]Ⅎ𝑥 𝑧𝐴)
6 bj-sbnf 32016 . . . . 5 ([𝑦 / 𝑧]Ⅎ𝑥 𝑧𝐴 ↔ Ⅎ𝑥[𝑦 / 𝑧]𝑧𝐴)
7 clelsb3 2716 . . . . . 6 ([𝑦 / 𝑧]𝑧𝐴𝑦𝐴)
87nfbii 1770 . . . . 5 (Ⅎ𝑥[𝑦 / 𝑧]𝑧𝐴 ↔ Ⅎ𝑥 𝑦𝐴)
96, 8bitri 263 . . . 4 ([𝑦 / 𝑧]Ⅎ𝑥 𝑧𝐴 ↔ Ⅎ𝑥 𝑦𝐴)
109albii 1737 . . 3 (∀𝑦[𝑦 / 𝑧]Ⅎ𝑥 𝑧𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
115, 10bitri 263 . 2 (∀𝑧𝑥 𝑧𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
121, 11bitri 263 1 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ∀wal 1473  Ⅎwnf 1699  [wsb 1867   ∈ wcel 1977  Ⅎwnfc 2738 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-cleq 2603  df-clel 2606  df-nfc 2740 This theorem is referenced by: (None)
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