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Mirrors > Home > ILE Home > Th. List > pm11.53 | GIF version |
Description: Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) |
Ref | Expression |
---|---|
pm11.53 | ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.21v 1884 | . . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓)) | |
2 | 1 | albii 1481 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑥(𝜑 → ∀𝑦𝜓)) |
3 | ax-17 1537 | . . . 4 ⊢ (𝜓 → ∀𝑥𝜓) | |
4 | 3 | hbal 1488 | . . 3 ⊢ (∀𝑦𝜓 → ∀𝑥∀𝑦𝜓) |
5 | 4 | 19.23h 1509 | . 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)) |
6 | 2, 5 | bitri 184 | 1 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1362 ∃wex 1503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie2 1505 ax-4 1521 ax-17 1537 ax-ial 1545 ax-i5r 1546 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: (None) |
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