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Theorem bj-nfcf 33497
 Description: Version of df-nfc 2921 with a disjoint variable 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 2921 . 2 (𝑥𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)
2 bj-nfcf.nf . . . . . 6 𝑦𝐴
32nfcri 2929 . . . . 5 𝑦 𝑧𝐴
43nfnf 2302 . . . 4 𝑦𝑥 𝑧𝐴
54sb8 2501 . . 3 (∀𝑧𝑥 𝑧𝐴 ↔ ∀𝑦[𝑦 / 𝑧]Ⅎ𝑥 𝑧𝐴)
6 bj-sbnf 33407 . . . . 5 ([𝑦 / 𝑧]Ⅎ𝑥 𝑧𝐴 ↔ Ⅎ𝑥[𝑦 / 𝑧]𝑧𝐴)
7 clelsb3 2888 . . . . . 6 ([𝑦 / 𝑧]𝑧𝐴𝑦𝐴)
87nfbii 1896 . . . . 5 (Ⅎ𝑥[𝑦 / 𝑧]𝑧𝐴 ↔ Ⅎ𝑥 𝑦𝐴)
96, 8bitri 267 . . . 4 ([𝑦 / 𝑧]Ⅎ𝑥 𝑧𝐴 ↔ Ⅎ𝑥 𝑦𝐴)
109albii 1863 . . 3 (∀𝑦[𝑦 / 𝑧]Ⅎ𝑥 𝑧𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
115, 10bitri 267 . 2 (∀𝑧𝑥 𝑧𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
121, 11bitri 267 1 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198  ∀wal 1599  Ⅎwnf 1827  [wsb 2011   ∈ wcel 2107  Ⅎwnfc 2919 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clel 2774  df-nfc 2921 This theorem is referenced by: (None)
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