Theorem ralrimi 2696

Description: Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 10-Oct-1999.)

Hypotheses

Ref	Expression
ralrimi.1	⊢ Ⅎxφ
ralrimi.2	⊢ (φ → (x ∈ A → ψ))

Assertion

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

Proof of Theorem ralrimi

Step	Hyp	Ref	Expression
1		ralrimi.1	. . 3 ⊢ Ⅎxφ
2		ralrimi.2	. . 3 ⊢ (φ → (x ∈ A → ψ))
3	1, 2	alrimi 1765	. 2 ⊢ (φ → ∀x(x ∈ A → ψ))
4		df-ral 2620	. 2 ⊢ (∀x ∈ A ψ ↔ ∀x(x ∈ A → ψ))
5	3, 4	sylibr 203	1 ⊢ (φ → ∀x ∈ A ψ)

Syntax hints:   → wi 4  ∀wal 1540  Ⅎwnf 1544   ∈ wcel 1710  ∀wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by:  ralrimiv  2697  reximdai  2723  r19.12  2728  rexlimd  2736  rexlimd2  2737  r19.37  2761  ralcom2  2776  2rmorex  3041  iineq2d  3990  ncfinraise  4482  mpteq2da  5667