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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 2368 and r19.12sn 4642. (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 2155 | . . 3 ⊢ Ⅎ𝑦∀𝑦𝜑 | |
2 | 1 | nfex 2343 | . 2 ⊢ Ⅎ𝑦∃𝑥∀𝑦𝜑 |
3 | sp 2182 | . . 3 ⊢ (∀𝑦𝜑 → 𝜑) | |
4 | 3 | eximi 1835 | . 2 ⊢ (∃𝑥∀𝑦𝜑 → ∃𝑥𝜑) |
5 | 2, 4 | alrimi 2213 | 1 ⊢ (∃𝑥∀𝑦𝜑 → ∀𝑦∃𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 ∃wex 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-11 2161 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-or 844 df-ex 1781 df-nf 1785 |
This theorem is referenced by: nfald 2347 bj-nfald 34445 pm11.61 40815 |
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