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Theorem elvd 2735
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.)
Hypothesis
Ref Expression
elvd.1 ((𝜑𝑥 ∈ V) → 𝜓)
Assertion
Ref Expression
elvd (𝜑𝜓)

Proof of Theorem elvd
StepHypRef Expression
1 vex 2733 . 2 𝑥 ∈ V
2 elvd.1 . 2 ((𝜑𝑥 ∈ V) → 𝜓)
31, 2mpan2 423 1 (𝜑𝜓)
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
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