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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2exbi | Structured version Visualization version GIF version |
Description: Theorem *11.341 in [WhiteheadRussell] p. 162. Theorem 19.18 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.) |
Ref | Expression |
---|---|
2exbi | ⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exbi 1852 | . . 3 ⊢ (∀𝑦(𝜑 ↔ 𝜓) → (∃𝑦𝜑 ↔ ∃𝑦𝜓)) | |
2 | 1 | alimi 1817 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → ∀𝑥(∃𝑦𝜑 ↔ ∃𝑦𝜓)) |
3 | exbi 1852 | . 2 ⊢ (∀𝑥(∃𝑦𝜑 ↔ ∃𝑦𝜓) → (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓)) | |
4 | 2, 3 | syl 17 | 1 ⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) → (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1539 ∃wex 1785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 |
This theorem depends on definitions: df-bi 206 df-ex 1786 |
This theorem is referenced by: (None) |
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