Theorem gen21 45356

Description: Virtual deduction generalizing rule for one quantifying variables and two virtual hypothesis. gen21 45356 is alrimdv 1958 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 45322	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	alrimdv 1958	. 2 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
4	3	dfvd2ir 45323	1 ⊢ (   𝜑   ,   𝜓   ▶   ∀𝑥𝜒   )

This proof depends on syntax axioms:  ∀wal 1567  (   wvd2 45314

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939

This proof depends on definitions:  df-bi 210  df-an 401  df-vd2 45315

This theorem is used by:  truniALTVD  45614  trintALTVD  45616  onfrALTlem2VD  45625