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Mirrors > Home > MPE Home > Th. List > Mathboxes > 19.33-2 | Structured version Visualization version GIF version |
Description: Theorem *11.421 in [WhiteheadRussell] p. 163. Theorem 19.33 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
Ref | Expression |
---|---|
19.33-2 | ⊢ ((∀𝑥∀𝑦𝜑 ∨ ∀𝑥∀𝑦𝜓) → ∀𝑥∀𝑦(𝜑 ∨ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orc 867 | . . 3 ⊢ (𝜑 → (𝜑 ∨ 𝜓)) | |
2 | 1 | 2alimi 1820 | . 2 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦(𝜑 ∨ 𝜓)) |
3 | olc 868 | . . 3 ⊢ (𝜓 → (𝜑 ∨ 𝜓)) | |
4 | 3 | 2alimi 1820 | . 2 ⊢ (∀𝑥∀𝑦𝜓 → ∀𝑥∀𝑦(𝜑 ∨ 𝜓)) |
5 | 2, 4 | jaoi 857 | 1 ⊢ ((∀𝑥∀𝑦𝜑 ∨ ∀𝑥∀𝑦𝜓) → ∀𝑥∀𝑦(𝜑 ∨ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 847 ∀wal 1541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 |
This theorem depends on definitions: df-bi 210 df-or 848 |
This theorem is referenced by: (None) |
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