Theorem el3v 36532

Description: New way (elv 3447, 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 3445	. 2 ⊢ 𝑥 ∈ V
2		vex 3445	. 2 ⊢ 𝑦 ∈ V
3		vex 3445	. 2 ⊢ 𝑧 ∈ V
4		el3v.1	. 2 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V) → 𝜑)
5	1, 2, 3, 4	mp3an 1460	1 ⊢ 𝜑

Syntax hints:   → wi 4   ∧ w3a 1086   ∈ wcel 2105  Vcvv 3441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443

This theorem is referenced by:  dfxrn2  36694