Theorem ralrimd 2988

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (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 2122	. 2 ⊢ (𝜑 → (𝜓 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜒)))
5		df-ral 2946	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒))
6	4, 5	syl6ibr 242	1 ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4  ∀wal 1521  Ⅎwnf 1748   ∈ wcel 2030  ∀wral 2941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-12 2087

This theorem depends on definitions:  df-bi 197  df-ex 1745  df-nf 1750  df-ral 2946

This theorem is referenced by:  reusv2lem3  4901  fliftfun  6602  mapxpen  8167  domtriomlem  9302  dedekind  10238  fzrevral  12463  matunitlindflem2  33536  riotasv3d  34564  ssralv2  39054  setrec1lem2  42760