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Mirrors > Home > ILE Home > Th. List > hbal | GIF version |
Description: If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
hbal.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
Ref | Expression |
---|---|
hbal | ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbal.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | 1 | alimi 1448 | . 2 ⊢ (∀𝑦𝜑 → ∀𝑦∀𝑥𝜑) |
3 | ax-7 1441 | . 2 ⊢ (∀𝑦∀𝑥𝜑 → ∀𝑥∀𝑦𝜑) | |
4 | 2, 3 | syl 14 | 1 ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-5 1440 ax-7 1441 ax-gen 1442 |
This theorem is referenced by: hba2 1544 aaanh 1579 hbex 1629 pm11.53 1888 euf 2024 hbral 2499 |
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