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Mirrors > Home > ILE Home > Th. List > 19.29r2 | GIF version |
Description: Variation of Theorem 19.29 of [Margaris] p. 90 with double quantification. (Contributed by NM, 3-Feb-2005.) |
Ref | Expression |
---|---|
19.29r2 | ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.29r 1512 | . 2 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓) → ∃𝑥(∃𝑦𝜑 ∧ ∀𝑦𝜓)) | |
2 | 19.29r 1512 | . . 3 ⊢ ((∃𝑦𝜑 ∧ ∀𝑦𝜓) → ∃𝑦(𝜑 ∧ 𝜓)) | |
3 | 2 | eximi 1491 | . 2 ⊢ (∃𝑥(∃𝑦𝜑 ∧ ∀𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓)) |
4 | 1, 3 | syl 14 | 1 ⊢ ((∃𝑥∃𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1241 ∃wex 1381 |
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-ie1 1382 ax-ie2 1383 ax-4 1400 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: (None) |
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