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Theorem bj-nfcf 34327
Description: Version of df-nfc 2962 with a disjoint variable condition replaced with a nonfreeness 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 2962 . 2 (𝑥𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)
2 bj-nfcf.nf . . . . . 6 𝑦𝐴
32nfcri 2967 . . . . 5 𝑦 𝑧𝐴
43nfnf 2346 . . . 4 𝑦𝑥 𝑧𝐴
54sb8v 2374 . . 3 (∀𝑧𝑥 𝑧𝐴 ↔ ∀𝑦[𝑦 / 𝑧]Ⅎ𝑥 𝑧𝐴)
6 bj-sbnf 34240 . . . . 5 ([𝑦 / 𝑧]Ⅎ𝑥 𝑧𝐴 ↔ Ⅎ𝑥[𝑦 / 𝑧]𝑧𝐴)
7 clelsb3 2941 . . . . . 6 ([𝑦 / 𝑧]𝑧𝐴𝑦𝐴)
87nfbii 1853 . . . . 5 (Ⅎ𝑥[𝑦 / 𝑧]𝑧𝐴 ↔ Ⅎ𝑥 𝑦𝐴)
96, 8bitri 278 . . . 4 ([𝑦 / 𝑧]Ⅎ𝑥 𝑧𝐴 ↔ Ⅎ𝑥 𝑦𝐴)
109albii 1821 . . 3 (∀𝑦[𝑦 / 𝑧]Ⅎ𝑥 𝑧𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
115, 10bitri 278 . 2 (∀𝑧𝑥 𝑧𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
121, 11bitri 278 1 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wal 1536  wnf 1785  [wsb 2069  wcel 2114  wnfc 2960
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-10 2145  ax-11 2161  ax-12 2178
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clel 2894  df-nfc 2962
This theorem is referenced by: (None)
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