Theorem gen12 44589

Description: Virtual deduction generalizing rule for two quantifying variables and one virtual hypothesis. gen12 44589 is alrimivv 1927 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 44542	. . 3 ⊢ (𝜑 → 𝜓)
3	2	alrimivv 1927	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦𝜓)
4	3	dfvd1ir 44544	1 ⊢ (   𝜑   ▶   ∀𝑥∀𝑦𝜓   )

Syntax hints:  ∀wal 1535  (   wvd1 44540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909

This theorem depends on definitions:  df-bi 207  df-vd1 44541

This theorem is referenced by:  sspwtr  44792  pwtrVD  44795  pwtrrVD  44796  suctrALT2VD  44807  truniALTVD  44849  trintALTVD  44851  suctrALTcfVD  44894