Theorem nfcvf 2817

Description: If 𝑥 and 𝑦 are distinct, then 𝑥 is not free in 𝑦. (Contributed by Mario Carneiro, 8-Oct-2016.)

Assertion

Ref	Expression
nfcvf	⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)

Proof of Theorem nfcvf

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2793	. 2 ⊢ Ⅎ𝑥𝑧
2		nfcv 2793	. 2 ⊢ Ⅎ𝑧𝑦
3		id 22	. 2 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦)
4	1, 2, 3	dvelimc 2816	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1521  Ⅎwnfc 2780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-cleq 2644  df-clel 2647  df-nfc 2782

This theorem is referenced by:  nfcvf2  2818  nfrald  2973  ralcom2  3133  nfreud  3141  nfrmod  3142  nfrmo  3144  nfdisj  4664  nfcvb  4928  nfriotad  6659  nfixp  7969  axextnd  9451  axrepndlem2  9453  axrepnd  9454  axunndlem1  9455  axunnd  9456  axpowndlem2  9458  axpowndlem4  9460  axregndlem2  9463  axregnd  9464  axinfndlem1  9465  axinfnd  9466  axacndlem4  9470  axacndlem5  9471  axacnd  9472  axextdist  31829  bj-nfcsym  33011