Theorem el3v 3465

Description: If a proposition is implied by 𝑥 ∈ V, 𝑦 ∈ V and 𝑧 ∈ V (which is true, see vex 3461), then it is true. Inference forms (with ⊢ 𝐴 ∈ V, ⊢ 𝐵 ∈ V and ⊢ 𝐶 ∈ V hypotheses) of the general theorems (proving ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → assertions) may be superfluous. (Contributed by Peter Mazsa, 13-Oct-2018.)

Hypothesis

Ref	Expression
el3v.1	⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝜑)

Assertion

Ref	Expression
el3v	⊢ 𝜑

Proof of Theorem el3v

Step	Hyp	Ref	Expression
1		vex 3461	. 2 ⊢ 𝑥 ∈ V
2		vex 3461	. 2 ⊢ 𝑦 ∈ V
3		vex 3461	. 2 ⊢ 𝑧 ∈ V
4		el3v.1	. 2 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝜑)
5	1, 2, 3, 4	mp3an 1490	1 ⊢ 𝜑

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   ∈ 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-3an 1105  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:  dfxrn2  39092  usgrgrtrirex  48773