Theorem e23 42375

Description: A virtual deduction elimination rule (see syl10 79). (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
e23.1	⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
e23.2	⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜏   )
e23.3	⊢ (𝜒 → (𝜏 → 𝜂))

Assertion

Ref	Expression
e23	⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜂   )

Proof of Theorem e23

Step	Hyp	Ref	Expression
1		e23.1	. . 3 ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
2	1	vd23 42222	. 2 ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜒   )
3		e23.2	. 2 ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜏   )
4		e23.3	. 2 ⊢ (𝜒 → (𝜏 → 𝜂))
5	2, 3, 4	e33 42354	1 ⊢ (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜂   )

Syntax hints:   → wi 4  (   wvd2 42197  (   wvd3 42207

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-vd2 42198  df-vd3 42210

This theorem is referenced by:  e23an  42376  suctrALT2VD  42456  rspsbc2VD  42475  tratrbVD  42481  imbi12VD  42493  imbi13VD  42494