Theorem nfcriOLDOLD 2897

Description: Obsolete version of nfcri 2893 as of 26-May-2024. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2139, ax-11 2156. (Revised by Gino Giotto, 23-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
nfcrii.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfcriOLDOLD	⊢ Ⅎ𝑥 𝑦 ∈ 𝐴

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem nfcriOLDOLD

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2821	. . 3 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	nfbidv 1926	. 2 ⊢ (𝑧 = 𝑦 → (Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐴))
3		nfcrii.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4		df-nfc 2888	. . . . 5 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴)
5	4	biimpi 215	. . . 4 ⊢ (Ⅎ𝑥𝐴 → ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴)
6		df-nf 1788	. . . . . 6 ⊢ (Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ (∃𝑥 𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴))
7	6	albii 1823	. . . . 5 ⊢ (∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ ∀𝑧(∃𝑥 𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴))
8		sp 2178	. . . . 5 ⊢ (∀𝑧(∃𝑥 𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴) → (∃𝑥 𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴))
9	7, 8	sylbi 216	. . . 4 ⊢ (∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴 → (∃𝑥 𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴))
10	3, 5, 9	mp2b 10	. . 3 ⊢ (∃𝑥 𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴)
11	10	nfi 1792	. 2 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
12	2, 11	chvarvv 2003	1 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1783  Ⅎwnf 1787   ∈ wcel 2108  Ⅎwnfc 2886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-nf 1788  df-clel 2817  df-nfc 2888

This theorem is referenced by: (None)