Theorem nfcvf2 2938

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 2373. See nfcv 2908 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 2937	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝑥)
2	1	naecoms 2430	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1539  Ⅎwnfc 2888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-13 2373

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-ex 1786  df-nf 1790  df-nfc 2890

This theorem is referenced by:  dfid3  5491  oprabid  7300  axrepndlem1  10332  axrepndlem2  10333  axrepnd  10334  axunnd  10336  axpowndlem3  10339  axpowndlem4  10340  axpownd  10341  axregndlem2  10343  axinfndlem1  10345  axinfnd  10346  axacndlem4  10350  axacndlem5  10351  axacnd  10352  bj-nfcsym  35063