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Mirrors > Home > MPE Home > Th. List > Mathboxes > hbimtg | Structured version Visualization version GIF version |
Description: A more general and closed form of hbim 2112. (Contributed by Scott Fenton, 13-Dec-2010.) |
Ref | Expression |
---|---|
hbimtg | ⊢ ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → ((𝜒 → 𝜓) → ∀𝑥(𝜑 → 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbntg 30955 | . . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜒) → (¬ 𝜒 → ∀𝑥 ¬ 𝜑)) | |
2 | pm2.21 119 | . . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜃)) | |
3 | 2 | alimi 1730 | . . . 4 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝜃)) |
4 | 1, 3 | syl6 34 | . . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜒) → (¬ 𝜒 → ∀𝑥(𝜑 → 𝜃))) |
5 | 4 | adantr 480 | . 2 ⊢ ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → (¬ 𝜒 → ∀𝑥(𝜑 → 𝜃))) |
6 | ala1 1755 | . . . 4 ⊢ (∀𝑥𝜃 → ∀𝑥(𝜑 → 𝜃)) | |
7 | 6 | imim2i 16 | . . 3 ⊢ ((𝜓 → ∀𝑥𝜃) → (𝜓 → ∀𝑥(𝜑 → 𝜃))) |
8 | 7 | adantl 481 | . 2 ⊢ ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → (𝜓 → ∀𝑥(𝜑 → 𝜃))) |
9 | 5, 8 | jad 173 | 1 ⊢ ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → ((𝜒 → 𝜓) → ∀𝑥(𝜑 → 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: hbimg 30959 |
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