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Theorem nfnfc 2770
 Description: Hypothesis builder for Ⅎ𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-13 2245. (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 2750 . 2 (𝑦𝐴 ↔ ∀𝑧𝑦 𝑧𝐴)
2 nfnfc.1 . . . . 5 𝑥𝐴
3 nfcr 2753 . . . . 5 (𝑥𝐴 → Ⅎ𝑥 𝑧𝐴)
42, 3ax-mp 5 . . . 4 𝑥 𝑧𝐴
54nfnf 2155 . . 3 𝑥𝑦 𝑧𝐴
65nfal 2150 . 2 𝑥𝑧𝑦 𝑧𝐴
71, 6nfxfr 1776 1 𝑥𝑦𝐴
 Colors of variables: wff setvar class Syntax hints:  ∀wal 1478  Ⅎwnf 1705   ∈ wcel 1987  Ⅎwnfc 2748 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-nfc 2750 This theorem is referenced by: (None)
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