Theorem el3v3 3460

Description: If a proposition is implied by 𝑧 ∈ V (which is true, see vex 3455) and two other antecedents, then it is implied by these other antecedents. (Contributed by Peter Mazsa, 16-Oct-2020.)

Hypothesis

Ref	Expression
el3v3.1	⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ V) → 𝜃)

Assertion

Ref	Expression
el3v3	⊢ ((𝜑 ∧ 𝜓) → 𝜃)

Proof of Theorem el3v3

Step	Hyp	Ref	Expression
1		vex 3455	. 2 ⊢ 𝑧 ∈ V
2		el3v3.1	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ V) → 𝜃)
3	1, 2	mp3an3 1479	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ 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:  el3v13  39145  el3v23  39146  dfgrlic2  49075  dfgrlic3  49077