Theorem nfcriiOLD 2900

Description: Obsolete version of nfcrii 2899 as of 23-May-2024. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nfcriOLD.1	⊢ Ⅎ𝑥𝐴

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem nfcriiOLD

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcriOLD.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
2	1	nfcri 2894	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
3	2	nfsbv 2324	. . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑧]𝑧 ∈ 𝐴
4	3	nf5ri 2188	. 2 ⊢ ([𝑦 / 𝑧]𝑧 ∈ 𝐴 → ∀𝑥[𝑦 / 𝑧]𝑧 ∈ 𝐴)
5		clelsb1 2866	. 2 ⊢ ([𝑦 / 𝑧]𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
6	5	albii 1822	. 2 ⊢ (∀𝑥[𝑦 / 𝑧]𝑧 ∈ 𝐴 ↔ ∀𝑥 𝑦 ∈ 𝐴)
7	4, 5, 6	3imtr3i 291	1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1537  [wsb 2067   ∈ 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  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-nf 1787  df-sb 2068  df-clel 2816  df-nfc 2889

This theorem is referenced by: (None)