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Mirrors > Home > ILE Home > Th. List > a5i | GIF version |
Description: Inference generalizing a consequent. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
a5i.1 | ⊢ (∀𝑥𝜑 → 𝜓) |
Ref | Expression |
---|---|
a5i | ⊢ (∀𝑥𝜑 → ∀𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hba1 1551 | . . 3 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | |
2 | ax-5 1458 | . . 3 ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥∀𝑥𝜑 → ∀𝑥𝜓)) | |
3 | 1, 2 | syl5 32 | . 2 ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) |
4 | a5i.1 | . 2 ⊢ (∀𝑥𝜑 → 𝜓) | |
5 | 3, 4 | mpg 1462 | 1 ⊢ (∀𝑥𝜑 → ∀𝑥𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-5 1458 ax-gen 1460 ax-ial 1545 |
This theorem is referenced by: hbae 1729 equveli 1770 hbsb2a 1817 hbsb2e 1818 aev 1823 dveeq2or 1827 hbsb2 1847 nfsb2or 1848 reu6 2941 |
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