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Mirrors > Home > MPE Home > Th. List > exan | Structured version Visualization version GIF version |
Description: Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) Reduce axiom dependencies. (Revised by BJ, 7-Jul-2021.) (Proof shortened by Wolf Lammen, 8-Oct-2021.) |
Ref | Expression |
---|---|
exan.1 | ⊢ (∃𝑥𝜑 ∧ 𝜓) |
Ref | Expression |
---|---|
exan | ⊢ ∃𝑥(𝜑 ∧ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exan.1 | . . 3 ⊢ (∃𝑥𝜑 ∧ 𝜓) | |
2 | 1 | simpli 474 | . 2 ⊢ ∃𝑥𝜑 |
3 | 1 | simpri 478 | . . . 4 ⊢ 𝜓 |
4 | pm3.21 464 | . . . 4 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓))) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (𝜑 → (𝜑 ∧ 𝜓)) |
6 | 5 | eximi 1762 | . 2 ⊢ (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)) |
7 | 2, 6 | ax-mp 5 | 1 ⊢ ∃𝑥(𝜑 ∧ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∃wex 1704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 |
This theorem is referenced by: bm1.3ii 4784 ac6s6f 33981 fnchoice 39188 |
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