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Mirrors > Home > MPE Home > Th. List > Mathboxes > el2v | Structured version Visualization version GIF version |
Description: New way (elv 33983, el2v 33984 theorems and el3v 33987 theorems) to shorten some proofs. Inference forms (with 𝐴 ∈ V and 𝐵 ∈ V hypotheses) of the general theorems (proving (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) →) may be superfluous. (Contributed by Peter Mazsa, 13-Oct-2018.) |
Ref | Expression |
---|---|
el2v.1 | ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝜑) |
Ref | Expression |
---|---|
el2v | ⊢ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3203 | . 2 ⊢ 𝑥 ∈ V | |
2 | vex 3203 | . 2 ⊢ 𝑦 ∈ V | |
3 | el2v.1 | . 2 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → 𝜑) | |
4 | 1, 2, 3 | mp2an 708 | 1 ⊢ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 Vcvv 3200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-12 2047 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-an 386 df-tru 1486 df-ex 1705 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-v 3202 |
This theorem is referenced by: vvdifopab 34024 inxprnres 34060 ineccnvmo 34122 alrmomo 34123 |
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