Theorem el123 42384

Description: A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
el123.1	⊢ (   𝜑   ▶   𝜓   )
el123.2	⊢ (   𝜒   ▶   𝜃   )
el123.3	⊢ (   𝜏   ▶   𝜂   )
el123.4	⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁)

Assertion

Ref	Expression
el123	⊢ (   (   𝜑   ,   𝜒   ,   𝜏   )   ▶   𝜁   )

Proof of Theorem el123

Step	Hyp	Ref	Expression
1		el123.1	. . . 4 ⊢ (   𝜑   ▶   𝜓   )
2	1	in1 42191	. . 3 ⊢ (𝜑 → 𝜓)
3		el123.2	. . . 4 ⊢ (   𝜒   ▶   𝜃   )
4	3	in1 42191	. . 3 ⊢ (𝜒 → 𝜃)
5		el123.3	. . . 4 ⊢ (   𝜏   ▶   𝜂   )
6	5	in1 42191	. . 3 ⊢ (𝜏 → 𝜂)
7		el123.4	. . 3 ⊢ ((𝜓 ∧ 𝜃 ∧ 𝜂) → 𝜁)
8	2, 4, 6, 7	syl3an 1159	. 2 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜁)
9	8	dfvd3anir 42216	1 ⊢ (   (   𝜑   ,   𝜒   ,   𝜏   )   ▶   𝜁   )

Syntax hints:   → wi 4   ∧ w3a 1086  (   wvd1 42189  (   wvhc3 42208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-vd1 42190  df-vhc3 42209

This theorem is referenced by:  suctrALTcfVD  42543