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Mirrors > Home > MPE Home > Th. List > 19.29 | Structured version Visualization version GIF version |
Description: Theorem 19.29 of [Margaris] p. 90. See also 19.29r 1878. (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 | pm3.2 471 | . . 3 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓))) | |
2 | 1 | aleximi 1835 | . 2 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑 ∧ 𝜓))) |
3 | 2 | imp 408 | 1 ⊢ ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∀wal 1540 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 |
This theorem is referenced by: 19.29x 1880 supsrlem 11054 1stccnp 22829 iscmet3 24673 isch3 30225 bnj849 33577 lfuhgr3 33753 axc11n11r 35177 bj-19.42t 35267 stoweidlem35 44350 |
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