Theorem el3v 3496

Description: If a proposition is implied by 𝑥 ∈ V, 𝑦 ∈ V and 𝑧 ∈ V (which is true, see vex 3492), 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 3492	. 2 ⊢ 𝑥 ∈ V
2		vex 3492	. 2 ⊢ 𝑦 ∈ V
3		vex 3492	. 2 ⊢ 𝑧 ∈ V
4		el3v.1	. 2 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝜑)
5	1, 2, 3, 4	mp3an 1461	1 ⊢ 𝜑

Syntax hints:   → wi 4   ∧ w3a 1087   ∈ wcel 2108  Vcvv 3488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490

This theorem is referenced by:  dfxrn2  38332  usgrgrtrirex  47799