Theorem ralimdv 2538

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 8-Oct-2003.)

Hypothesis

Ref	Expression
ralimdv.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
ralimdv	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem ralimdv

Step	Hyp	Ref	Expression
1		ralimdv.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	adantr 274	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒))
3	2	ralimdva 2537	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∈ wcel 2141  ∀wral 2448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-4 1503  ax-17 1519

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-ral 2453

This theorem is referenced by:  poss  4281  sess1  4320  sess2  4321  riinint  4870  dffo4  5642  dffo5  5643  isoini2  5796  rdgivallem  6358  iinerm  6583  xpf1o  6820  exmidontriimlem3  7193  exmidontriim  7195  resqrexlemgt0  10977  cau3lem  11071  caubnd2  11074  climshftlemg  11258  climcau  11303  climcaucn  11307  serf0  11308  modfsummodlemstep  11413  bezoutlemmain  11946  ctinf  12378  strsetsid  12442  fiinbas  12806  baspartn  12807  lmtopcnp  13009  rescncf  13327  limcresi  13394