Theorem nfcvf 2511

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 2489	. 2 ⊢ Ⅎxz
2		nfcv 2489	. 2 ⊢ Ⅎzy
3		id 19	. 2 ⊢ (z = y → z = y)
4	1, 2, 3	dvelimc 2510	1 ⊢ (¬ ∀x x = y → Ⅎxy)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540   = wceq 1642  Ⅎwnfc 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478

This theorem is referenced by:  nfcvf2  2512  nfrald  2665  ralcom2  2775  nfreud  2783  nfrmod  2784  nfrmo  2786  nfiotad  4342