Theorem gen12 43369

Description: Virtual deduction generalizing rule for two quantifying variables and one virtual hypothesis. gen12 43369 is alrimivv 1931 with virtual deductions. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
gen12.1	⊢ (   𝜑   ▶   𝜓   )

Assertion

Ref	Expression
gen12	⊢ (   𝜑   ▶   ∀𝑥∀𝑦𝜓   )

Distinct variable groups:   𝜑,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑦)

Proof of Theorem gen12

Step	Hyp	Ref	Expression
1		gen12.1	. . . 4 ⊢ (   𝜑   ▶   𝜓   )
2	1	in1 43322	. . 3 ⊢ (𝜑 → 𝜓)
3	2	alrimivv 1931	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦𝜓)
4	3	dfvd1ir 43324	1 ⊢ (   𝜑   ▶   ∀𝑥∀𝑦𝜓   )

Syntax hints:  ∀wal 1539  (   wvd1 43320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913

This theorem depends on definitions:  df-bi 206  df-vd1 43321

This theorem is referenced by:  sspwtr  43572  pwtrVD  43575  pwtrrVD  43576  suctrALT2VD  43587  truniALTVD  43629  trintALTVD  43631  suctrALTcfVD  43674