Theorem nfcvf 2370

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 2347	. 2 ⊢ Ⅎ𝑥𝑧
2		nfcv 2347	. 2 ⊢ Ⅎ𝑧𝑦
3		id 19	. 2 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦)
4	1, 2, 3	dvelimc 2369	1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1370  Ⅎwnfc 2334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-cleq 2197  df-clel 2200  df-nfc 2336

This theorem is referenced by:  nfcvf2  2371