Theorem el2v1 38938

Description: New way (elv 3462, 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 3461	. 2 ⊢ 𝑥 ∈ V
2		el2v1.1	. 2 ⊢ ((𝑥 ∈ V ∧ 𝜑) → 𝜓)
3	1, 2	mpan 703	1 ⊢ (𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2146  Vcvv 3457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459

This theorem is used by:  el3v12  38941  el3v13  38942  exan3  39009  exanres3  39011  ecin0  39061  disjsuc2  39123  dfpre3  39187  exeupre  39200  preuniqval  39205  eldm1cossres2  39260  brcosscnv  39271  eqvrelqsel  39409  dfpetparts2  39681