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Mirrors > Home > MPE Home > Th. List > Mathboxes > aaanv | Structured version Visualization version GIF version |
Description: Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 2156. (Contributed by Andrew Salmon, 24-May-2011.) |
Ref | Expression |
---|---|
aaanv | ⊢ ((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥∀𝑦(𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1830 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1830 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | 1, 2 | aaan 2156 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) |
4 | 3 | bicomi 213 | 1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥∀𝑦(𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ 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-11 2021 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-nf 1701 |
This theorem is referenced by: (None) |
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