Theorem elvd 2731

Description: Technical lemma used to shorten proofs. If a proposition is implied by x e. _V (which is true, see vex 2729) and another antecedent, then it is implied by the other antecedent. (Contributed by Peter Mazsa, 23-Oct-2018.)

Hypothesis

Ref	Expression
elvd.1	|- ( ( ph /\ x e. _V ) -> ps )

Assertion

Ref	Expression
elvd	|- ( ph -> ps )

Proof of Theorem elvd

Step	Hyp	Ref	Expression
1		vex 2729	. 2 |- x e. _V
2		elvd.1	. 2 |- ( ( ph /\ x e. _V ) -> ps )
3	1, 2	mpan2 422	1 |- ( ph -> ps )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2136   _Vcvv 2726

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728

This theorem is referenced by:  omp1eomlem  7059  imasnopn  12939  pw1nct  13883