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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 3222 | . 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑 | |
2 | 1 | nf5ri 2194 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1534 ∀wral 3141 |
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-10 2144 ax-12 2176 |
This theorem depends on definitions: df-bi 209 df-or 844 df-ex 1780 df-nf 1784 df-ral 3146 |
This theorem is referenced by: bnj1095 32057 bnj1309 32298 mpobi123f 35444 hbra2VD 41200 tratrbVD 41201 ssralv2VD 41206 |
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