Theorem ralrimi 2548

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 1522	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
4		df-ral 2460	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
5	3, 4	sylibr 134	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

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:  ralrimiv  2549  reximdai  2575  r19.12  2583  rexlimd  2591  rexlimd2  2592  r19.29af2  2617  r19.37  2629  ralidm  3524  iineq2d  3907  mpteq2da  4093  onintonm  4517  mpteqb  5607  fmptdf  5674  eusvobj2  5861  tfri3  6368  mapxpen  6848  fodjuomnilemdc  7142  cc3  7267  fimaxre2  11235  fprodcllemf  11621  fprodap0f  11644  fprodle  11648  zsupcllemstep  11946  bezoutlemmain  11999  bezoutlemzz  12003  exmidunben  12427  mulcncf  14094  limccnp2lem  14148