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Mirrors > Home > MPE Home > Th. List > Mathboxes > 19.36vv | Structured version Visualization version GIF version |
Description: Theorem *11.43 in [WhiteheadRussell] p. 163. Theorem 19.36 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
19.36vv | ⊢ (∃𝑥∃𝑦(𝜑 → 𝜓) ↔ (∀𝑥∀𝑦𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.36v 1891 | . . 3 ⊢ (∃𝑦(𝜑 → 𝜓) ↔ (∀𝑦𝜑 → 𝜓)) | |
2 | 1 | exbii 1764 | . 2 ⊢ (∃𝑥∃𝑦(𝜑 → 𝜓) ↔ ∃𝑥(∀𝑦𝜑 → 𝜓)) |
3 | 19.36v 1891 | . 2 ⊢ (∃𝑥(∀𝑦𝜑 → 𝜓) ↔ (∀𝑥∀𝑦𝜑 → 𝜓)) | |
4 | 2, 3 | bitri 263 | 1 ⊢ (∃𝑥∃𝑦(𝜑 → 𝜓) ↔ (∀𝑥∀𝑦𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 ∃wex 1695 |
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 |
This theorem depends on definitions: df-bi 196 df-ex 1696 |
This theorem is referenced by: (None) |
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