Theorem el3v2 39164

Description: New way (elv 3455, and the theorems beginning with "el2v" or "el3v") to shorten some proofs. (Contributed by Peter Mazsa, 16-Oct-2020.)

Hypothesis

Ref	Expression
el3v2.1	⊢ ((𝜑 ∧ 𝑦 ∈ V ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
el3v2	⊢ ((𝜑 ∧ 𝜒) → 𝜃)

Proof of Theorem el3v2

Step	Hyp	Ref	Expression
1		vex 3454	. 2 ⊢ 𝑦 ∈ V
2		el3v2.1	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ V ∧ 𝜒) → 𝜃)
3	1, 2	mp3an2 1478	1 ⊢ ((𝜑 ∧ 𝜒) → 𝜃)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2144  Vcvv 3450

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452

This theorem is used by: (None)