Theorem gen11 39662

Description: Virtual deduction generalizing rule for one quantifying variable and one virtual hypothesis. alrimiv 2026 is gen11 39662 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
gen11.1	⊢ (   𝜑   ▶   𝜓   )

Assertion

Ref	Expression
gen11	⊢ (   𝜑   ▶   ∀𝑥𝜓   )

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem gen11

Step	Hyp	Ref	Expression
1		gen11.1	. . . 4 ⊢ (   𝜑   ▶   𝜓   )
2		dfvd1imp 39612	. . . 4 ⊢ ((   𝜑   ▶   𝜓   ) → (𝜑 → 𝜓))
3	1, 2	ax-mp 5	. . 3 ⊢ (𝜑 → 𝜓)
4	3	alrimiv 2026	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
5		dfvd1impr 39613	. 2 ⊢ ((𝜑 → ∀𝑥𝜓) → (   𝜑   ▶   ∀𝑥𝜓   ))
6	4, 5	ax-mp 5	1 ⊢ (   𝜑   ▶   ∀𝑥𝜓   )

Syntax hints:   → wi 4  ∀wal 1654  (   wvd1 39606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009

This theorem depends on definitions:  df-bi 199  df-vd1 39607

This theorem is referenced by:  trsspwALT  39865  snssiALTVD  39874  sstrALT2VD  39881  elex2VD  39885  elex22VD  39886  tpid3gVD  39889  trsbcVD  39924  sbcssgVD  39930  csbingVD  39931  onfrALTVD  39938  csbsngVD  39940  csbxpgVD  39941  csbrngVD  39943  csbunigVD  39945  csbfv12gALTVD  39946  ax6e2eqVD  39954  ax6e2ndeqVD  39956  sspwimpVD  39966  sspwimpcfVD  39968