Theorem gen21 37865

Description: Virtual deduction generalizing rule for one quantifying variables and two virtual hypothesis. gen21 37865 is alrimdv 1844 with virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
gen21.1	⊢ (   𝜑   ,   𝜓   ▶   𝜒   )

Assertion

Ref	Expression
gen21	⊢ (   𝜑   ,   𝜓   ▶   ∀𝑥𝜒   )

Distinct variable groups:   𝜑,𝑥   𝜓,𝑥

Allowed substitution hint:   𝜒(𝑥)

Proof of Theorem gen21

Step	Hyp	Ref	Expression
1		gen21.1	. . . 4 ⊢ (   𝜑   ,   𝜓   ▶   𝜒   )
2	1	dfvd2i 37822	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	alrimdv 1844	. 2 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
4	3	dfvd2ir 37823	1 ⊢ (   𝜑   ,   𝜓   ▶   ∀𝑥𝜒   )

Syntax hints:  ∀wal 1473  (   wvd2 37814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827

This theorem depends on definitions:  df-bi 196  df-an 385  df-vd2 37815

This theorem is referenced by:  truniALTVD  38136  trintALTVD  38138  onfrALTlem2VD  38147