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Theorem nfcriOLD 2970
 Description: Obsolete version of nfcri 2967 as of 3-Jun-2024. (Contributed by Mario Carneiro, 11-Aug-2016.) Avoid ax-10 2145, ax-11 2161. (Revised by Gino Giotto, 23-May-2024.) Avoid ax-12 2178. (Revised by SN, 26-May-2024.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
nfcrii.1 𝑥𝐴
Assertion
Ref Expression
nfcriOLD 𝑥 𝑦𝐴
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.
StepHypRef Expression
1 eleq1w 2896 . . 3 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
21nfbidv 1923 . 2 (𝑧 = 𝑦 → (Ⅎ𝑥 𝑧𝐴 ↔ Ⅎ𝑥 𝑦𝐴))
3 nfcrii.1 . . . 4 𝑥𝐴
4 df-nfc 2962 . . . . 5 (𝑥𝐴 ↔ ∀𝑧𝑥 𝑧𝐴)
54biimpi 219 . . . 4 (𝑥𝐴 → ∀𝑧𝑥 𝑧𝐴)
6 df-nf 1786 . . . . . 6 (Ⅎ𝑥 𝑧𝐴 ↔ (∃𝑥 𝑧𝐴 → ∀𝑥 𝑧𝐴))
76albii 1821 . . . . 5 (∀𝑧𝑥 𝑧𝐴 ↔ ∀𝑧(∃𝑥 𝑧𝐴 → ∀𝑥 𝑧𝐴))
8 eleq1w 2896 . . . . . . . 8 (𝑧 = 𝑤 → (𝑧𝐴𝑤𝐴))
98exbidv 1922 . . . . . . 7 (𝑧 = 𝑤 → (∃𝑥 𝑧𝐴 ↔ ∃𝑥 𝑤𝐴))
108albidv 1921 . . . . . . 7 (𝑧 = 𝑤 → (∀𝑥 𝑧𝐴 ↔ ∀𝑥 𝑤𝐴))
119, 10imbi12d 348 . . . . . 6 (𝑧 = 𝑤 → ((∃𝑥 𝑧𝐴 → ∀𝑥 𝑧𝐴) ↔ (∃𝑥 𝑤𝐴 → ∀𝑥 𝑤𝐴)))
1211spw 2041 . . . . 5 (∀𝑧(∃𝑥 𝑧𝐴 → ∀𝑥 𝑧𝐴) → (∃𝑥 𝑧𝐴 → ∀𝑥 𝑧𝐴))
137, 12sylbi 220 . . . 4 (∀𝑧𝑥 𝑧𝐴 → (∃𝑥 𝑧𝐴 → ∀𝑥 𝑧𝐴))
143, 5, 13mp2b 10 . . 3 (∃𝑥 𝑧𝐴 → ∀𝑥 𝑧𝐴)
1514nfi 1790 . 2 𝑥 𝑧𝐴
162, 15chvarvv 2005 1 𝑥 𝑦𝐴
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536  ∃wex 1781  Ⅎwnf 1785   ∈ wcel 2114  Ⅎwnfc 2960 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-clel 2894  df-nfc 2962 This theorem is referenced by: (None)
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