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Mirrors > Home > MPE Home > Th. List > 19.29r | Structured version Visualization version GIF version |
Description: Variation of 19.29 1874. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2020.) |
Ref | Expression |
---|---|
19.29r | ⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.21 475 | . . 3 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓))) | |
2 | 1 | aleximi 1833 | . 2 ⊢ (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))) |
3 | 2 | impcom 411 | 1 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1536 ∃wex 1781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 |
This theorem is referenced by: 19.29r2 1876 19.29x 1877 intab 4868 imadif 6408 kmlem6 9566 hashgt23el 13781 2ndcdisj 22061 fmcncfil 31284 bnj907 32349 funen1cnv 32467 loop1cycl 32497 umgr2cycl 32501 bj-19.41al 34105 |
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