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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 1812. See alimdh 1819, alimdv 1917 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 2193 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) |
3 | alimd.2 | . 2 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
4 | 2, 3 | alimdh 1819 | 1 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1536 Ⅎwnf 1785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-ex 1782 df-nf 1786 |
This theorem is referenced by: alrimdd 2212 nfald 2336 mo3 2623 2mo 2710 axpowndlem3 10010 axextbdist 33158 pm11.71 41101 |
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