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| Mirrors > Home > MPE Home > Th. List > Mathboxes > el2v1 | Structured version Visualization version GIF version | ||
| Description: New way (elv 3485, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 23-Oct-2018.) | 
| Ref | Expression | 
|---|---|
| el2v1.1 | ⊢ ((𝑥 ∈ V ∧ 𝜑) → 𝜓) | 
| Ref | Expression | 
|---|---|
| el2v1 | ⊢ (𝜑 → 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | vex 3484 | . 2 ⊢ 𝑥 ∈ V | |
| 2 | el2v1.1 | . 2 ⊢ ((𝑥 ∈ V ∧ 𝜑) → 𝜓) | |
| 3 | 1, 2 | mpan 690 | 1 ⊢ (𝜑 → 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 | 
| This theorem is referenced by: el3v12 38227 el3v13 38228 exan3 38295 exanres3 38297 ecin0 38353 disjsuc2 38392 eldm1cossres2 38462 brcosscnv 38473 eqvrelqsel 38617 | 
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