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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 | pm3.21 474 | . . 3 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓))) | |
2 | 1 | aleximi 1832 | . 2 ⊢ (∀𝑥𝜓 → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))) |
3 | 2 | impcom 410 | 1 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 |
This theorem is referenced by: 19.29r2 1876 19.29x 1877 intab 4908 imadif 6440 kmlem6 9583 hashgt23el 13788 2ndcdisj 22066 fmcncfil 31176 bnj907 32241 funen1cnv 32359 loop1cycl 32386 umgr2cycl 32390 bj-19.41al 33994 |
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