Theorem ralrimi 2577

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 1545	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
4		df-ral 2489	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
5	3, 4	sylibr 134	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4  ∀wal 1371  Ⅎwnf 1483   ∈ wcel 2176  ∀wral 2484

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 1470  ax-gen 1472  ax-4 1533

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-ral 2489

This theorem is referenced by:  ralrimiv  2578  reximdai  2604  r19.12  2612  rexlimd  2620  rexlimd2  2621  r19.29af2  2646  r19.37  2658  ralidm  3561  iineq2d  3947  mpteq2da  4134  onintonm  4566  mpteqb  5672  fmptdf  5739  eusvobj2  5932  tfri3  6455  mapxpen  6947  fodjuomnilemdc  7248  cc3  7382  zsupcllemstep  10374  fimaxre2  11571  fprodcllemf  11957  fprodap0f  11980  fprodle  11984  bezoutlemmain  12352  bezoutlemzz  12356  exmidunben  12830  mulcncf  15113  limccnp2lem  15181