Theorem ralrimd 3245

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) For a version based on fewer axioms see ralrimdv 3138. (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 2227	. 2 ⊢ (𝜑 → (𝜓 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜒)))
5		df-ral 3055	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒))
6	4, 5	imbitrrdi 253	1 ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4  ∀wal 1545  Ⅎwnf 1790   ∈ wcel 2119  ∀wral 3054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-ex 1787  df-nf 1791  df-ral 3055

This theorem is referenced by:  reusv2lem3  5336  fliftfun  7263  mapxpen  9078  domtriomlem  10362  dedekind  11307  fzrevral  13564  matunitlindflem2  37991  riotasv3d  39459  ssralv2  44982  setrec1lem2  50185