Theorem gen12 42127

Description: Virtual deduction generalizing rule for two quantifying variables and one virtual hypothesis. gen12 42127 is alrimivv 1932 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 42080	. . 3 ⊢ (𝜑 → 𝜓)
3	2	alrimivv 1932	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦𝜓)
4	3	dfvd1ir 42082	1 ⊢ (   𝜑   ▶   ∀𝑥∀𝑦𝜓   )

Syntax hints:  ∀wal 1537  (   wvd1 42078

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-vd1 42079

This theorem is referenced by:  sspwtr  42330  pwtrVD  42333  pwtrrVD  42334  suctrALT2VD  42345  truniALTVD  42387  trintALTVD  42389  suctrALTcfVD  42432