Theorem gen21 45200

Description: Virtual deduction generalizing rule for one quantifying variables and two virtual hypothesis. gen21 45200 is alrimdv 1951 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 45166	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	alrimdv 1951	. 2 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
4	3	dfvd2ir 45167	1 ⊢ (   𝜑   ,   𝜓   ▶   ∀𝑥𝜒   )

Syntax hints:  ∀wal 1560  (   wvd2 45158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932

This theorem depends on definitions:  df-bi 209  df-an 400  df-vd2 45159

This theorem is referenced by:  truniALTVD  45458  trintALTVD  45460  onfrALTlem2VD  45469