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Mirrors > Home > MPE Home > Th. List > Mathboxes > pm11.52 | Structured version Visualization version GIF version |
Description: Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) |
Ref | Expression |
---|---|
pm11.52 | ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ¬ ∀𝑥∀𝑦(𝜑 → ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-an 386 | . . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓)) | |
2 | 1 | 2exbii 1775 | . 2 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ∃𝑥∃𝑦 ¬ (𝜑 → ¬ 𝜓)) |
3 | 2nalexn 1755 | . 2 ⊢ (¬ ∀𝑥∀𝑦(𝜑 → ¬ 𝜓) ↔ ∃𝑥∃𝑦 ¬ (𝜑 → ¬ 𝜓)) | |
4 | 2, 3 | bitr4i 267 | 1 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ ¬ ∀𝑥∀𝑦(𝜑 → ¬ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∀wal 1481 ∃wex 1704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 |
This theorem is referenced by: (None) |
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