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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > el3v | Structured version Visualization version GIF version |
Description: New way (elv 3476, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. Inference forms (with ⊢ 𝐴 ∈ V, ⊢ 𝐵 ∈ V and ⊢ 𝐶 ∈ V hypotheses) of the general theorems (proving ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → assertions) may be superfluous. (Contributed by Peter Mazsa, 13-Oct-2018.) |
Ref | Expression |
---|---|
el3v.1 | ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝜑) |
Ref | Expression |
---|---|
el3v | ⊢ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3474 | . 2 ⊢ 𝑥 ∈ V | |
2 | vex 3474 | . 2 ⊢ 𝑦 ∈ V | |
3 | vex 3474 | . 2 ⊢ 𝑧 ∈ V | |
4 | el3v.1 | . 2 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝜑) | |
5 | 1, 2, 3, 4 | mp3an 1458 | 1 ⊢ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 ∈ wcel 2099 Vcvv 3470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-3an 1087 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3472 |
This theorem is referenced by: dfxrn2 37842 |
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