Theorem el2122old 42339

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

Hypotheses

Ref	Expression
el2122old.1	⊢ (   (   𝜑   ,   𝜓   )   ▶   𝜒   )
el2122old.2	⊢ (   𝜓   ▶   𝜃   )
el2122old.3	⊢ (   𝜓   ▶   𝜏   )
el2122old.4	⊢ ((𝜒 ∧ 𝜃 ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
el2122old	⊢ (   (   𝜑   ,   𝜓   )   ▶   𝜂   )

Proof of Theorem el2122old

Step	Hyp	Ref	Expression
1		el2122old.1	. . . 4 ⊢ (   (   𝜑   ,   𝜓   )   ▶   𝜒   )
2	1	dfvd2ani 42203	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
3		el2122old.2	. . . 4 ⊢ (   𝜓   ▶   𝜃   )
4	3	in1 42191	. . 3 ⊢ (𝜓 → 𝜃)
5		el2122old.3	. . . 4 ⊢ (   𝜓   ▶   𝜏   )
6	5	in1 42191	. . 3 ⊢ (𝜓 → 𝜏)
7		el2122old.4	. . 3 ⊢ ((𝜒 ∧ 𝜃 ∧ 𝜏) → 𝜂)
8	2, 4, 6, 7	eel2122old 42338	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜂)
9	8	dfvd2anir 42204	1 ⊢ (   (   𝜑   ,   𝜓   )   ▶   𝜂   )

Syntax hints:   → wi 4   ∧ w3a 1086  (   wvd1 42189  (   wvhc2 42200

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-vhc2 42201

This theorem is referenced by:  suctrALTcfVD  42543