Theorem ralrimd 3273

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) For a version based on fewer axioms see ralrimdv 3166. (Contributed by NM, 16-Feb-2004.)

Hypotheses

Ref	Expression
ralrimd.1	⊢ Ⅎ𝑥𝜑
ralrimd.2	⊢ Ⅎ𝑥𝜓
ralrimd.3	⊢ (𝜑 → (𝜓 → (𝑥 ∈ 𝐴 → 𝜒)))

Assertion

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

Proof of Theorem ralrimd

Step	Hyp	Ref	Expression
1		ralrimd.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		ralrimd.2	. . 3 ⊢ Ⅎ𝑥𝜓
3		ralrimd.3	. . 3 ⊢ (𝜑 → (𝜓 → (𝑥 ∈ 𝐴 → 𝜒)))
4	1, 2, 3	alrimd 2254	. 2 ⊢ (𝜑 → (𝜓 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜒)))
5		df-ral 3083	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒))
6	4, 5	imbitrrdi 255	1 ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 𝜒))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  Ⅎwnf 1816   ∈ wcel 2146  ∀wral 3082

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  ax-6 2000  ax-7 2041  ax-12 2216

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817  df-ral 3083

This theorem is used by:  reusv2lem3  5376  fliftfun  7321  mapxpen  9141  domtriomlem  10444  dedekind  11391  fzrevral  13659  matunitlindflem2  38309  riotasv3d  39775  ssralv2  45281  setrec1lem2  50507