Theorem gen12 45442

Description: Virtual deduction generalizing rule for two quantifying variables and one virtual hypothesis. gen12 45442 is alrimivv 1961 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 45395	. . 3 ⊢ (𝜑 → 𝜓)
3	2	alrimivv 1961	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦𝜓)
4	3	dfvd1ir 45397	1 ⊢ (   𝜑   ▶   ∀𝑥∀𝑦𝜓   )

This proof depends on syntax axioms:  ∀wal 1568  (   wvd1 45393

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-vd1 45394

This theorem is used by:  sspwtr  45644  pwtrVD  45647  pwtrrVD  45648  suctrALT2VD  45659  truniALTVD  45701  trintALTVD  45703  suctrALTcfVD  45746