Theorem gen11 39527

Description: Virtual deduction generalizing rule for one quantifying variable and one virtual hypothesis. alrimiv 2022 is gen11 39527 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 39477	. . . 4 ⊢ ((   𝜑   ▶   𝜓   ) → (𝜑 → 𝜓))
3	1, 2	ax-mp 5	. . 3 ⊢ (𝜑 → 𝜓)
4	3	alrimiv 2022	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
5		dfvd1impr 39478	. 2 ⊢ ((𝜑 → ∀𝑥𝜓) → (   𝜑   ▶   ∀𝑥𝜓   ))
6	4, 5	ax-mp 5	1 ⊢ (   𝜑   ▶   ∀𝑥𝜓   )

Syntax hints:   → wi 4  ∀wal 1650  (   wvd1 39471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005

This theorem depends on definitions:  df-bi 198  df-vd1 39472

This theorem is referenced by:  trsspwALT  39730  snssiALTVD  39739  sstrALT2VD  39746  elex2VD  39750  elex22VD  39751  tpid3gVD  39754  trsbcVD  39789  sbcssgVD  39795  csbingVD  39796  onfrALTVD  39803  csbsngVD  39805  csbxpgVD  39806  csbrngVD  39808  csbunigVD  39810  csbfv12gALTVD  39811  ax6e2eqVD  39819  ax6e2ndeqVD  39821  sspwimpVD  39831  sspwimpcfVD  39833