Theorem nfnfc 2346

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 2328	. 2 ⊢ (Ⅎ𝑦𝐴 ↔ ∀𝑧Ⅎ𝑦 𝑧 ∈ 𝐴)
2		nfnfc.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2333	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
4	3	nfnf 1591	. . 3 ⊢ Ⅎ𝑥Ⅎ𝑦 𝑧 ∈ 𝐴
5	4	nfal 1590	. 2 ⊢ Ⅎ𝑥∀𝑧Ⅎ𝑦 𝑧 ∈ 𝐴
6	1, 5	nfxfr 1488	1 ⊢ Ⅎ𝑥Ⅎ𝑦𝐴

Syntax hints:  ∀wal 1362  Ⅎwnf 1474   ∈ wcel 2167  Ⅎwnfc 2326

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-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-cleq 2189  df-clel 2192  df-nfc 2328

This theorem is referenced by: (None)