Theorem ralrimi 2621

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 1575	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
4		df-ral 2533	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
5	3, 4	sylibr 134	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1400  Ⅎwnf 1513   ∈ wcel 2209  ∀wral 2528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563

This proof depends on definitions:  df-bi 117  df-nf 1514  df-ral 2533

This theorem is used by:  ralrimiv  2622  reximdai  2648  r19.12  2657  rexlimd  2665  rexlimd2  2666  r19.29af2  2691  r19.37  2703  ralidm  3628  iineq2d  4032  mpteq2da  4220  onintonm  4664  mpteqb  5796  fmptdf  5865  eusvobj2  6071  funimass4f  6359  tfri3  6638  mapxpen  7148  fodjuomnilemdc  7484  cc3  7634  zsupcllemstep  10664  fimaxre2  11995  fprodcllemf  12382  fprodap0f  12405  fprodle  12409  bezoutlemmain  12777  bezoutlemzz  12781  exmidunben  13319  mulcncf  15711  limccnp2lem  15779  lfgrnloopen  16386