Theorem nfcvf2 2958

Description: If 𝑥 and 𝑦 are distinct, then 𝑦 is not free in 𝑥. Usage of this theorem is discouraged because it depends on ax-13 2410. See nfcv 2931 for a version that replaces the distinctor with a disjoint variable condition, requiring fewer axioms. (Contributed by Mario Carneiro, 5-Dec-2016.) (New usage is discouraged.)

Assertion

Ref	Expression
nfcvf2	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥)

Proof of Theorem nfcvf2

Step	Hyp	Ref	Expression
1		nfcvf 2957	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝑥)
2	1	naecoms 2467	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1565  Ⅎwnfc 2916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-13 2410

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-nfc 2918

This theorem is referenced by:  dfid3  5560  oprabid  7443  axrepndlem1  10576  axrepndlem2  10577  axrepnd  10578  axunnd  10580  axpowndlem3  10583  axpowndlem4  10584  axpownd  10585  axregndlem2  10587  axinfndlem1  10589  axinfnd  10590  axacndlem4  10594  axacndlem5  10595  axacnd  10596  bj-nfcsym  37422