Theorem el3v12 39164

Description: New way (elv 3456, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 11-Jul-2021.)

Hypothesis

Ref	Expression
el3v12.1	⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
el3v12	⊢ (𝜒 → 𝜃)

Proof of Theorem el3v12

Step	Hyp	Ref	Expression
1		el3v12.1	. . 3 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝜒) → 𝜃)
2	1	el3v1 39162	. 2 ⊢ ((𝑦 ∈ V ∧ 𝜒) → 𝜃)
3	2	el2v1 39161	1 ⊢ (𝜒 → 𝜃)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   ∈ wcel 2145  Vcvv 3451

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 2147  ax-9 2155  ax-ext 2733

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 2740  df-cleq 2753  df-clel 2836  df-v 3453

This theorem is used by:  ralmo  39292