Theorem el2v1 38728

Description: New way (elv 3459, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 23-Oct-2018.)

Hypothesis

Ref	Expression
el2v1.1	⊢ ((𝑥 ∈ V ∧ 𝜑) → 𝜓)

Assertion

Ref	Expression
el2v1	⊢ (𝜑 → 𝜓)

Proof of Theorem el2v1

Step	Hyp	Ref	Expression
1		vex 3458	. 2 ⊢ 𝑥 ∈ V
2		el2v1.1	. 2 ⊢ ((𝑥 ∈ V ∧ 𝜑) → 𝜓)
3	1, 2	mpan 700	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2142  Vcvv 3454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456

This theorem is referenced by:  el3v12  38731  el3v13  38732  exan3  38799  exanres3  38801  ecin0  38851  disjsuc2  38913  dfpre3  38977  exeupre  38990  preuniqval  38995  eldm1cossres2  39050  brcosscnv  39061  eqvrelqsel  39199  dfpetparts2  39471