Theorem ralrimd 2555

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 1610	. 2 ⊢ (𝜑 → (𝜓 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜒)))
5		df-ral 2460	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒))
6	4, 5	syl6ibr 162	1 ⊢ (𝜑 → (𝜓 → ∀𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4  ∀wal 1351  Ⅎwnf 1460   ∈ wcel 2148  ∀wral 2455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-4 1510

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-ral 2460

This theorem is referenced by:  ralrimdv  2556  fliftfun  5794  mapxpen  6845  fzrevral  10100