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Mirrors > Home > MPE Home > Th. List > alimd | Structured version Visualization version GIF version |
Description: Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1813. See alimdh 1820, alimdv 1918 for variants requiring fewer axioms. (Contributed by Mario Carneiro, 24-Sep-2016.) |
Ref | Expression |
---|---|
alimd.1 | ⊢ Ⅎ𝑥𝜑 |
alimd.2 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
alimd | ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alimd.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | nf5ri 2194 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) |
3 | alimd.2 | . 2 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
4 | 2, 3 | alimdh 1820 | 1 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1537 Ⅎwnf 1786 |
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 1912 ax-6 1971 ax-7 2016 ax-12 2176 |
This theorem depends on definitions: df-bi 210 df-ex 1783 df-nf 1787 |
This theorem is referenced by: alrimdd 2213 nfald 2337 mo3 2583 2mo 2670 axpowndlem3 10044 axextbdist 33277 pm11.71 41459 |
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