Theorem elvd 2691

Description: Technical lemma used to shorten proofs. If a proposition is implied by 𝑥 ∈ V (which is true, see vex 2689) and another antecedent, then it is implied by the other antecedent. (Contributed by Peter Mazsa, 23-Oct-2018.)

Hypothesis

Ref	Expression
elvd.1	⊢ ((𝜑 ∧ 𝑥 ∈ V) → 𝜓)

Assertion

Ref	Expression
elvd	⊢ (𝜑 → 𝜓)

Proof of Theorem elvd

Step	Hyp	Ref	Expression
1		vex 2689	. 2 ⊢ 𝑥 ∈ V
2		elvd.1	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ V) → 𝜓)
3	1, 2	mpan2 421	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 1480  Vcvv 2686

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688

This theorem is referenced by:  omp1eomlem  6979  imasnopn  12468