Theorem el12 42346

Description: Virtual deduction form of syl2an 596. (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 42191	. . 3 ⊢ (𝜑 → 𝜓)
3		el12.2	. . . 4 ⊢ (   𝜏   ▶   𝜒   )
4	3	in1 42191	. . 3 ⊢ (𝜏 → 𝜒)
5		el12.3	. . 3 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)
6	2, 4, 5	syl2an 596	. 2 ⊢ ((𝜑 ∧ 𝜏) → 𝜃)
7	6	dfvd2anir 42204	1 ⊢ (   (   𝜑   ,   𝜏   )   ▶   𝜃   )

Syntax hints:   → wi 4   ∧ wa 396  (   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-vd1 42190  df-vhc2 42201

This theorem is referenced by:  elpwgdedVD  42537  sspwimpVD  42539  sspwimpcfVD  42541  suctrALTcfVD  42543