Theorem ralrimdv 2470

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 27-May-1998.)

Hypothesis

Ref	Expression
ralrimdv.1	⊢ (𝜑 → (𝜓 → (𝑥 ∈ 𝐴 → 𝜒)))

Assertion

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

Distinct variable groups:   𝜑,𝑥   𝜓,𝑥

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

Proof of Theorem ralrimdv

Step	Hyp	Ref	Expression
1		nfv 1476	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1476	. 2 ⊢ Ⅎ𝑥𝜓
3		ralrimdv.1	. 2 ⊢ (𝜑 → (𝜓 → (𝑥 ∈ 𝐴 → 𝜒)))
4	1, 2, 3	ralrimd 2469	1 ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ∈ wcel 1448  ∀wral 2375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-gen 1393  ax-4 1455  ax-17 1474

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-ral 2380

This theorem is referenced by:  ralrimdva  2471  ralrimivv  2472  nneneq  6680  fzrevral  9726  topbas  12018  neipsm  12105  cnpnei  12169  metcnp3  12435