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Theorem nfnfc 2967
 Description: Hypothesis builder for Ⅎ𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-13 2379. (Revised by Wolf Lammen, 10-Dec-2019.)
Hypothesis
Ref Expression
nfnfc.1 𝑥𝐴
Assertion
Ref Expression
nfnfc 𝑥𝑦𝐴

Proof of Theorem nfnfc
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-nfc 2938 . 2 (𝑦𝐴 ↔ ∀𝑧𝑦 𝑧𝐴)
2 nfnfc.1 . . . . . 6 𝑥𝐴
3 df-nfc 2938 . . . . . 6 (𝑥𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)
42, 3mpbi 233 . . . . 5 𝑧𝑥 𝑧𝐴
54spi 2181 . . . 4 𝑥 𝑧𝐴
65nfnf 2334 . . 3 𝑥𝑦 𝑧𝐴
76nfal 2331 . 2 𝑥𝑧𝑦 𝑧𝐴
81, 7nfxfr 1854 1 𝑥𝑦𝐴
 Colors of variables: wff setvar class Syntax hints:  ∀wal 1536  Ⅎwnf 1785   ∈ wcel 2111  Ⅎwnfc 2936 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-10 2142  ax-11 2158  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-nfc 2938 This theorem is referenced by: (None)
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