Theorem ralrimi 2601

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 1568	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
4		df-ral 2513	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
5	3, 4	sylibr 134	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4  ∀wal 1393  Ⅎwnf 1506   ∈ wcel 2200  ∀wral 2508

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 1493  ax-gen 1495  ax-4 1556

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-ral 2513

This theorem is referenced by:  ralrimiv  2602  reximdai  2628  r19.12  2637  rexlimd  2645  rexlimd2  2646  r19.29af2  2671  r19.37  2683  ralidm  3592  iineq2d  3985  mpteq2da  4173  onintonm  4609  mpteqb  5727  fmptdf  5794  eusvobj2  5993  tfri3  6519  mapxpen  7017  fodjuomnilemdc  7322  cc3  7465  zsupcllemstep  10461  fimaxre2  11754  fprodcllemf  12140  fprodap0f  12163  fprodle  12167  bezoutlemmain  12535  bezoutlemzz  12539  exmidunben  13013  mulcncf  15298  limccnp2lem  15366  lfgrnloopen  15947