Theorem nfcvf2 2927

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 2377. See nfcv 2899 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 2926	. 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝑥)
2	1	naecoms 2434	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  Ⅎwnfc 2884

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 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-nfc 2886

This theorem is referenced by:  dfid3  5523  oprabid  7393  axrepndlem1  10509  axrepndlem2  10510  axrepnd  10511  axunnd  10513  axpowndlem3  10516  axpowndlem4  10517  axpownd  10518  axregndlem2  10520  axinfndlem1  10522  axinfnd  10523  axacndlem4  10527  axacndlem5  10528  axacnd  10529  bj-nfcsym  37225