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Mirrors > Home > MPE Home > Th. List > 19.12 | Structured version Visualization version GIF version |
Description: Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! But sometimes the converse does hold, as in 19.12vv 2345 and r19.12sn 4656. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 3-Jan-2018.) |
Ref | Expression |
---|---|
19.12 | ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2148 | . . 3 ⊢ Ⅎ𝑦∀𝑦𝜑 | |
2 | 1 | nfex 2318 | . 2 ⊢ Ⅎ𝑦∃𝑥∀𝑦𝜑 |
3 | sp 2176 | . . 3 ⊢ (∀𝑦𝜑 → 𝜑) | |
4 | 3 | eximi 1837 | . 2 ⊢ (∃𝑥∀𝑦𝜑 → ∃𝑥𝜑) |
5 | 2, 4 | alrimi 2206 | 1 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1537 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-11 2154 ax-12 2171 |
This theorem depends on definitions: df-bi 206 df-or 845 df-ex 1783 df-nf 1787 |
This theorem is referenced by: nfald 2322 bj-nfald 35308 pm11.61 42011 |
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