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Metamath Proof Explorer

Theorem el3v

Description: If a proposition is implied by x e.V , y e. V and z e.V (which is true, see vex ), then it is true. Inference forms (with |- A e. V , |- B e.V and |- C e. V hypotheses) of the general theorems (proving |- ( ( A e. V /\ B e. W /\ C e. X ) -> assertions) may be superfluous. (Contributed by Peter Mazsa, 13-Oct-2018)

		Ref	Expression
	Hypothesis	el3v.1	⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → 𝜑 )
	Assertion	el3v	⊢ 𝜑

Proof

Step	Hyp	Ref	Expression
1		el3v.1	⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V ) → 𝜑 )
2		vex	⊢ 𝑥 ∈ V
3		vex	⊢ 𝑦 ∈ V
4		vex	⊢ 𝑧 ∈ V
5	2 3 4 1	mp3an	⊢ 𝜑