Theorem el3v3 3463

Description: If a proposition is implied by 𝑧 ∈ V (which is true, see vex 3458) 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 3458	. 2 ⊢ 𝑧 ∈ V
2		el3v3.1	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝑧 ∈ V) → 𝜃)
3	1, 2	mp3an3 1471	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1098   ∈ wcel 2142  Vcvv 3454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1100  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456

This theorem is referenced by:  el3v13  38729  el3v23  38730  dfgrlic2  48627  dfgrlic3  48629