Theorem el12 45548

Description: Virtual deduction form of syl2an 608. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
el12.1	⊢ (   𝜑   ▶   𝜓   )
el12.2	⊢ (   𝜏   ▶   𝜒   )
el12.3	⊢ ((𝜓 ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
el12	⊢ (   (   𝜑   ,   𝜏   )   ▶   𝜃   )

Proof of Theorem el12

Step	Hyp	Ref	Expression
1		el12.1	. . . 4 ⊢ (   𝜑   ▶   𝜓   )
2	1	in1 45394	. . 3 ⊢ (𝜑 → 𝜓)
3		el12.2	. . . 4 ⊢ (   𝜏   ▶   𝜒   )
4	3	in1 45394	. . 3 ⊢ (𝜏 → 𝜒)
5		el12.3	. . 3 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
6	2, 4, 5	syl2an 608	. 2 ⊢ ((𝜑 ∧ 𝜏) → 𝜃)
7	6	dfvd2anir 45407	1 ⊢ (   (   𝜑   ,   𝜏   )   ▶   𝜃   )

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  (   wvd1 45392  (   wvhc2 45403

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-vd1 45393  df-vhc2 45404

This theorem is used by:  elpwgdedVD  45739  sspwimpVD  45741  sspwimpcfVD  45743  suctrALTcfVD  45745