Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > elvd | GIF version |
Description: Technical lemma used to shorten proofs. If a proposition is implied by 𝑥 ∈ V (which is true, see vex 2733) and another antecedent, then it is implied by the other antecedent. (Contributed by Peter Mazsa, 23-Oct-2018.) |
Ref | Expression |
---|---|
elvd.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ V) → 𝜓) |
Ref | Expression |
---|---|
elvd | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2733 | . 2 ⊢ 𝑥 ∈ V | |
2 | elvd.1 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ V) → 𝜓) | |
3 | 1, 2 | mpan2 423 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2141 Vcvv 2730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-v 2732 |
This theorem is referenced by: omp1eomlem 7071 imasnopn 13093 pw1nct 14036 |
Copyright terms: Public domain | W3C validator |