Theorem nfcvf2 2919

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 2370. See nfcv 2891 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 2918	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝑥)
2	1	naecoms 2427	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1538  Ⅎwnfc 2876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-13 2370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-nfc 2878

This theorem is referenced by:  dfid3  5529  oprabid  7401  axrepndlem1  10521  axrepndlem2  10522  axrepnd  10523  axunnd  10525  axpowndlem3  10528  axpowndlem4  10529  axpownd  10530  axregndlem2  10532  axinfndlem1  10534  axinfnd  10535  axacndlem4  10539  axacndlem5  10540  axacnd  10541  bj-nfcsym  36860