Theorem nfnfc 2319

Description: Hypothesis builder for Ⅎ𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nfnfc.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfnfc	⊢ Ⅎ𝑥Ⅎ𝑦𝐴

Proof of Theorem nfnfc

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nfc 2301	. 2 ⊢ (Ⅎ𝑦𝐴 ↔ ∀𝑧Ⅎ𝑦 𝑧 ∈ 𝐴)
2		nfnfc.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2306	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
4	3	nfnf 1570	. . 3 ⊢ Ⅎ𝑥Ⅎ𝑦 𝑧 ∈ 𝐴
5	4	nfal 1569	. 2 ⊢ Ⅎ𝑥∀𝑧Ⅎ𝑦 𝑧 ∈ 𝐴
6	1, 5	nfxfr 1467	1 ⊢ Ⅎ𝑥Ⅎ𝑦𝐴

Syntax hints:  ∀wal 1346  Ⅎwnf 1453   ∈ wcel 2141  Ⅎwnfc 2299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301

This theorem is referenced by: (None)