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Theorem nfcriiOLD 2952
Description: Obsolete version of nfcrii 2951 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 𝑥𝐴
Assertion
Ref Expression
nfcriiOLD (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem nfcriiOLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfcriOLD.1 . . . . 5 𝑥𝐴
21nfcri 2946 . . . 4 𝑥 𝑧𝐴
32nfsbv 2341 . . 3 𝑥[𝑦 / 𝑧]𝑧𝐴
43nf5ri 2194 . 2 ([𝑦 / 𝑧]𝑧𝐴 → ∀𝑥[𝑦 / 𝑧]𝑧𝐴)
5 clelsb3 2920 . 2 ([𝑦 / 𝑧]𝑧𝐴𝑦𝐴)
65albii 1821 . 2 (∀𝑥[𝑦 / 𝑧]𝑧𝐴 ↔ ∀𝑥 𝑦𝐴)
74, 5, 63imtr3i 294 1 (𝑦𝐴 → ∀𝑥 𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1536  [wsb 2069  wcel 2112  wnfc 2939
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 2114  ax-10 2143  ax-11 2159  ax-12 2176
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-sb 2070  df-clel 2873  df-nfc 2941
This theorem is referenced by: (None)
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