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Mirrors > Home > ILE Home > Th. List > 19.12 | GIF version |
Description: Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
19.12 | ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hba1 1521 | . . 3 ⊢ (∀𝑦𝜑 → ∀𝑦∀𝑦𝜑) | |
2 | 1 | hbex 1616 | . 2 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥∀𝑦𝜑) |
3 | ax-4 1488 | . . 3 ⊢ (∀𝑦𝜑 → 𝜑) | |
4 | 3 | eximi 1580 | . 2 ⊢ (∃𝑥∀𝑦𝜑 → ∃𝑥𝜑) |
5 | 2, 4 | alrimih 1446 | 1 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1330 ∃wex 1469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-ial 1515 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: hbexd 1673 nfexd 1735 cbvexdh 1899 |
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