Theorem nfcrii 2928

Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nfcri.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfcrii	⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem nfcrii

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcri.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2	1	nfcriv 2925	. . 3 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
3	2	nf5ri 2179	. 2 ⊢ (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴)
4	3	hblem 2891	1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1599   ∈ wcel 2107  Ⅎwnfc 2919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clel 2774  df-nfc 2921

This theorem is referenced by:  nfcri  2929  cleqfOLD  2964  abeq2fOLD  2967  bnj1230  31472  bnj1000  31610  bnj1204  31679  bnj1307  31690  bnj1311  31691  bnj1398  31701  bnj1466  31720  bnj1467  31721  bnj1523  31738