Theorem ralrimivw 2698

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 18-Jun-2014.)

Hypothesis

Ref	Expression
ralrimivw.1	⊢ (φ → ψ)

Assertion

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

Distinct variable group:   φ,x

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

Proof of Theorem ralrimivw

Step	Hyp	Ref	Expression
1		ralrimivw.1	. . 3 ⊢ (φ → ψ)
2	1	a1d 22	. 2 ⊢ (φ → (x ∈ A → ψ))
3	2	ralrimiv 2696	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:  2rmorex  3040  riinrab  4041  nnadjoinpw  4521  dmxp  4923  eqfnfv  5392  mpteq12dv  5656  mpt2eq12  5662  clos1nrel  5886  uniqs  5984  enprmaplem5  6080  nchoicelem6  6294  dmfrec  6316