Theorem gen21 42128

Description: Virtual deduction generalizing rule for one quantifying variables and two virtual hypothesis. gen21 42128 is alrimdv 1933 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 42094	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	2	alrimdv 1933	. 2 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
4	3	dfvd2ir 42095	1 ⊢ (   𝜑   ,   𝜓   ▶   ∀𝑥𝜒   )

Syntax hints:  ∀wal 1537  (   wvd2 42086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914

This theorem depends on definitions:  df-bi 206  df-an 396  df-vd2 42087

This theorem is referenced by:  truniALTVD  42387  trintALTVD  42389  onfrALTlem2VD  42398