Theorem nfci 2783

Description: Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nfci.1	⊢ Ⅎ𝑥 𝑦 ∈ 𝐴

Assertion

Ref	Expression
nfci	⊢ Ⅎ𝑥𝐴

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem nfci

Step	Hyp	Ref	Expression
1		df-nfc 2782	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
2		nfci.1	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
3	1, 2	mpgbir 1766	1 ⊢ Ⅎ𝑥𝐴

Syntax hints:  Ⅎwnf 1748   ∈ wcel 2030  Ⅎwnfc 2780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762

This theorem depends on definitions:  df-bi 197  df-nfc 2782

This theorem is referenced by:  nfcii  2784  nfcv  2793  nfab1  2795  nfab  2798  fpwrelmap  29636  esumfzf  30259  bj-nfab1  32910  fsumiunss  40125  climsuse  40158  climinff  40161  fnlimfvre  40224  limsupre3uzlem  40285  pimdecfgtioc  41246  pimincfltioc  41247  smfmullem4  41322  smflimsupmpt  41356