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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > el2v1 | Structured version Visualization version GIF version |
Description: New way (elv 3413, 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 3411 | . 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 400 ∈ wcel 2112 Vcvv 3407 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-tru 1542 df-ex 1783 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-v 3409 |
This theorem is referenced by: el3v12 35920 el3v13 35921 exan3 35976 exanres3 35978 ecin0 36031 eldm1cossres2 36126 brcosscnv 36137 eqvrelqsel 36276 |
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