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Mirrors > Home > ILE Home > Th. List > aaanh | GIF version |
Description: Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.) |
Ref | Expression |
---|---|
aaanh.1 | ⊢ (𝜑 → ∀𝑦𝜑) |
aaanh.2 | ⊢ (𝜓 → ∀𝑥𝜓) |
Ref | Expression |
---|---|
aaanh | ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aaanh.1 | . . . 4 ⊢ (𝜑 → ∀𝑦𝜑) | |
2 | 1 | 19.28h 1549 | . . 3 ⊢ (∀𝑦(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑦𝜓)) |
3 | 2 | albii 1457 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ ∀𝑥(𝜑 ∧ ∀𝑦𝜓)) |
4 | aaanh.2 | . . . 4 ⊢ (𝜓 → ∀𝑥𝜓) | |
5 | 4 | hbal 1464 | . . 3 ⊢ (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓) |
6 | 5 | 19.27h 1547 | . 2 ⊢ (∀𝑥(𝜑 ∧ ∀𝑦𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) |
7 | 3, 6 | bitri 183 | 1 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1434 ax-7 1435 ax-gen 1436 ax-4 1497 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: mo23 2054 2eu4 2106 |
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