Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > alimdh | Structured version Visualization version GIF version |
Description: Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1817. (Contributed by NM, 4-Jan-2002.) |
Ref | Expression |
---|---|
alimdh.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
alimdh.2 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
alimdh | ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alimdh.1 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | alimdh.2 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
3 | 2 | al2imi 1822 | . 2 ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) |
4 | 1, 3 | syl 17 | 1 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-gen 1802 ax-4 1816 |
This theorem is referenced by: alrimdh 1870 alimdv 1923 hbald 2172 alimd 2209 bj-nfald 35304 dral1-o 36914 ax12indalem 36955 ax12inda2ALT 36956 |
Copyright terms: Public domain | W3C validator |