Theorem ralrimiv 2696

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.)

Hypothesis

Ref	Expression
ralrimiv.1	⊢ (φ → (x ∈ A → ψ))

Assertion

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

Distinct variable group:   φ,x

Allowed substitution hints:   ψ(x)   A(x)

Proof of Theorem ralrimiv

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎxφ
2		ralrimiv.1	. 2 ⊢ (φ → (x ∈ A → ψ))
3	1, 2	ralrimi 2695	1 ⊢ (φ → ∀x ∈ A ψ)

Syntax hints:   → wi 4   ∈ wcel 1710  ∀wral 2614

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2619

This theorem is referenced by:  ralrimiva  2697  ralrimivw  2698  ralrimivv  2705  r19.27av  2752  rr19.3v  2980  rabssdv  3346  rzal  3651  pw1disj  4167  nnadjoin  4520  fmpt2d  5695  nnc3n3p1  6278