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Mirrors > Home > MPE Home > Th. List > nfra2w | Structured version Visualization version GIF version |
Description: Similar to Lemma 24 of [Monk2] p. 114, except the quantification of the antecedent is restricted. Derived automatically from hbra2VD 41268. Version of nfra2 3227 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Alan Sare, 31-Dec-2011.) (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
nfra2w | ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2976 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | nfra1 3218 | . 2 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 𝜑 | |
3 | 1, 2 | nfralw 3224 | 1 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnf 1783 ∀wral 3137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 |
This theorem is referenced by: invdisj 5043 reusv3 5299 dedekind 10796 dedekindle 10797 mreexexd 16914 gsummatr01lem4 21262 ordtconnlem1 31188 bnj1379 32123 tratrb 40944 islptre 41974 sprsymrelfo 43733 |
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