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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 1520 | . . 3 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | |
2 | ax-5 1423 | . . 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 1427 | 1 ⊢ (∀𝑥𝜑 → ∀𝑥𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1329 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-5 1423 ax-gen 1425 ax-ial 1514 |
This theorem is referenced by: hbae 1696 equveli 1732 hbsb2a 1778 hbsb2e 1779 aev 1784 dveeq2or 1788 hbsb2 1808 nfsb2or 1809 reu6 2873 |
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