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Mirrors > Home > ILE Home > Th. List > 19.21-2 | GIF version |
Description: Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.) |
Ref | Expression |
---|---|
19.21-2.1 | ⊢ Ⅎ𝑥𝜑 |
19.21-2.2 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
19.21-2 | ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥∀𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.21-2.2 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | 19.21 1475 | . . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓)) |
3 | 2 | albii 1359 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑥(𝜑 → ∀𝑦𝜓)) |
4 | 19.21-2.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
5 | 4 | 19.21 1475 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) ↔ (𝜑 → ∀𝑥∀𝑦𝜓)) |
6 | 3, 5 | bitri 173 | 1 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥∀𝑦𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1241 Ⅎwnf 1349 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-4 1400 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-nf 1350 |
This theorem is referenced by: (None) |
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