Theorem nfcriOLD 2897

Description: Obsolete version of nfcri 2894 as of 3-Jun-2024. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2137, ax-11 2154. (Revised by Gino Giotto, 23-May-2024.) Avoid ax-12 2171. (Revised by SN, 26-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
nfcrii.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfcriOLD	⊢ Ⅎ𝑥 𝑦 ∈ 𝐴

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem nfcriOLD

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2821	. . 3 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	nfbidv 1925	. 2 ⊢ (𝑧 = 𝑦 → (Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐴))
3		nfcrii.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4		df-nfc 2889	. . . . 5 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴)
5	4	biimpi 215	. . . 4 ⊢ (Ⅎ𝑥𝐴 → ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴)
6		df-nf 1787	. . . . . 6 ⊢ (Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ (∃𝑥 𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴))
7	6	albii 1822	. . . . 5 ⊢ (∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴 ↔ ∀𝑧(∃𝑥 𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴))
8		eleq1w 2821	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
9	8	exbidv 1924	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (∃𝑥 𝑧 ∈ 𝐴 ↔ ∃𝑥 𝑤 ∈ 𝐴))
10	8	albidv 1923	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (∀𝑥 𝑧 ∈ 𝐴 ↔ ∀𝑥 𝑤 ∈ 𝐴))
11	9, 10	imbi12d 345	. . . . . 6 ⊢ (𝑧 = 𝑤 → ((∃𝑥 𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴) ↔ (∃𝑥 𝑤 ∈ 𝐴 → ∀𝑥 𝑤 ∈ 𝐴)))
12	11	spw 2037	. . . . 5 ⊢ (∀𝑧(∃𝑥 𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴) → (∃𝑥 𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴))
13	7, 12	sylbi 216	. . . 4 ⊢ (∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴 → (∃𝑥 𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴))
14	3, 5, 13	mp2b 10	. . 3 ⊢ (∃𝑥 𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴)
15	14	nfi 1791	. 2 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
16	2, 15	chvarvv 2002	1 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴

Syntax hints:   → wi 4  ∀wal 1537  ∃wex 1782  Ⅎwnf 1786   ∈ wcel 2106  Ⅎwnfc 2887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-nf 1787  df-clel 2816  df-nfc 2889

This theorem is referenced by: (None)