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| Description: Theorem 19.29 of [Margaris] p. 90. See also 19.29r 1874. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.) | 
| Ref | Expression | 
|---|---|
| 19.29 | ⊢ ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | pm3.2 469 | . . 3 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓))) | |
| 2 | 1 | aleximi 1832 | . 2 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑 ∧ 𝜓))) | 
| 3 | 2 | imp 406 | 1 ⊢ ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 ∃wex 1779 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 | 
| This theorem is referenced by: 19.29x 1876 supsrlem 11151 1stccnp 23470 iscmet3 25327 isch3 31260 bnj849 34939 lfuhgr3 35125 axc11n11r 36684 bj-19.42t 36774 stoweidlem35 46050 | 
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