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Mirrors > Home > MPE Home > Th. List > 2moex | Structured version Visualization version GIF version |
Description: Double quantification with "at most one". Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker 2moexv 2628 when possible. (Contributed by NM, 3-Dec-2001.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2moex | ⊢ (∃*𝑥∃𝑦𝜑 → ∀𝑦∃*𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfe1 2153 | . . 3 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
2 | 1 | nfmo 2561 | . 2 ⊢ Ⅎ𝑦∃*𝑥∃𝑦𝜑 |
3 | 19.8a 2180 | . . 3 ⊢ (𝜑 → ∃𝑦𝜑) | |
4 | 3 | moimi 2544 | . 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 1541 ∃wex 1787 ∃*wmo 2537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-10 2143 ax-11 2160 ax-12 2177 ax-13 2371 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-mo 2539 |
This theorem is referenced by: 2eu2 2653 |
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