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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, 6-Nov-2022.) Expand hypothesis. (Revised by Steven Nguyen, 19-Jun-2023.) |
Ref | Expression |
---|---|
exan.1 | ⊢ ∃𝑥𝜑 |
exan.2 | ⊢ 𝜓 |
Ref | Expression |
---|---|
exan | ⊢ ∃𝑥(𝜑 ∧ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exan.1 | . 2 ⊢ ∃𝑥𝜑 | |
2 | exan.2 | . . 3 ⊢ 𝜓 | |
3 | 2 | jctr 525 | . 2 ⊢ (𝜑 → (𝜑 ∧ 𝜓)) |
4 | 1, 3 | eximii 1839 | 1 ⊢ ∃𝑥(𝜑 ∧ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∃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 397 df-ex 1783 |
This theorem is referenced by: bm1.3ii 5224 ac6s6f 36339 fnchoice 42553 |
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