Theorem nfcvf 2925

Description: If 𝑥 and 𝑦 are distinct, then 𝑥 is not free in 𝑦. Usage of this theorem is discouraged because it depends on ax-13 2376. See nfcv 2898 for a version that replaces the distinctor with a disjoint variable condition, requiring fewer axioms. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-ext 2708. (Revised by Wolf Lammen, 10-May-2023.) (New usage is discouraged.)

Assertion

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

Proof of Theorem nfcvf

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1916	. 2 ⊢ Ⅎ𝑤 ¬ ∀𝑥 𝑥 = 𝑦
2		nfv 1916	. . 3 ⊢ Ⅎ𝑥 𝑤 ∈ 𝑧
3		elequ2 2129	. . 3 ⊢ (𝑧 = 𝑦 → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝑦))
4	2, 3	dvelimnf 2457	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑤 ∈ 𝑦)
5	1, 4	nfcd 2891	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  Ⅎwnfc 2883

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 2376

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 2885

This theorem is referenced by:  nfcvf2  2926  nfrald  3334  ralcom2  3339  nfrmod  3385  nfreud  3386  nfrmo  3387  nfdisj  5065  nfcvb  5318  nfriotad  7335  nfixp  8865  axextnd  10514  axrepndlem2  10516  axrepnd  10517  axunndlem1  10518  axunnd  10519  axpowndlem2  10521  axpowndlem4  10523  axregndlem2  10526  axregnd  10527  axinfndlem1  10528  axinfnd  10529  axacndlem4  10533  axacndlem5  10534  axacnd  10535  axsepg2  35225  axsepg2ALT  35226  axnulg  35251  axextdist  35979  bj-nfcsym  37206