Theorem elvd 2735

Description: Technical lemma used to shorten proofs. If a proposition is implied by x e. _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	|- ( ( ph /\ x e. _V ) -> ps )

Assertion

Ref	Expression
elvd	|- ( ph -> ps )

Proof of Theorem elvd

Step	Hyp	Ref	Expression
1		vex 2733	. 2 |- x e. _V
2		elvd.1	. 2 |- ( ( ph /\ x e. _V ) -> ps )
3	1, 2	mpan2 423	1 |- ( ph -> ps )

Syntax hints:   -> wi 4   /\ wa 103   e. 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