Theorem ralrimiv 2697

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 2696	1 ⊢ (φ → ∀x ∈ A ψ)

Syntax hints:   → wi 4   ∈ wcel 1710  ∀wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2620

This theorem is referenced by:  ralrimiva  2698  ralrimivw  2699  ralrimivv  2706  r19.27av  2753  rr19.3v  2981  rabssdv  3347  rzal  3652  pw1disj  4168  nnadjoin  4521  fmpt2d  5696  nnc3n3p1  6279