![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > hbra1 | Structured version Visualization version GIF version |
Description: The setvar 𝑥 is not free in ∀𝑥 ∈ 𝐴𝜑. (Contributed by NM, 18-Oct-1996.) (Proof shortened by Wolf Lammen, 7-Dec-2019.) |
Ref | Expression |
---|---|
hbra1 | ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥∀𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfra1 3266 | . 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑 | |
2 | 1 | nf5ri 2189 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 ∀wral 3061 |
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 1914 ax-6 1972 ax-7 2012 ax-10 2138 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-or 847 df-ex 1783 df-nf 1787 df-ral 3062 |
This theorem is referenced by: bnj1095 33450 bnj1309 33691 mpobi123f 36667 hbra2VD 43230 tratrbVD 43231 ssralv2VD 43236 |
Copyright terms: Public domain | W3C validator |