Theorem elvd 2744

Description: Technical lemma used to shorten proofs. If a proposition is implied by 𝑥 ∈ V (which is true, see vex 2742) 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 2742	. 2 ⊢ 𝑥 ∈ V
2		elvd.1	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ V) → 𝜓)
3	1, 2	mpan2 425	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2148  Vcvv 2739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2741

This theorem is referenced by:  omp1eomlem  7095  subrgpropd  13374  imasnopn  13884  pw1nct  14837