Theorem el3v 35651

Description: New way (elv 3446, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. 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 3444	. 2 ⊢ 𝑥 ∈ V
2		vex 3444	. 2 ⊢ 𝑦 ∈ V
3		vex 3444	. 2 ⊢ 𝑧 ∈ V
4		el3v.1	. 2 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝜑)
5	1, 2, 3, 4	mp3an 1458	1 ⊢ 𝜑

Syntax hints:   → wi 4   ∧ w3a 1084   ∈ wcel 2111  Vcvv 3441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443

This theorem is referenced by:  dfxrn2  35788