Theorem ralimdv 2694

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 8-Oct-2003.)

Hypothesis

Ref	Expression
ralimdv.1	⊢ (φ → (ψ → χ))

Assertion

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

Distinct variable group:   φ,x

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

Proof of Theorem ralimdv

Step	Hyp	Ref	Expression
1		ralimdv.1	. . 3 ⊢ (φ → (ψ → χ))
2	1	adantr 451	. 2 ⊢ ((φ ∧ x ∈ A) → (ψ → χ))
3	2	ralimdva 2693	1 ⊢ (φ → (∀x ∈ A ψ → ∀x ∈ A χ))

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-an 360  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by:  dffo4  5424  dffo5  5425  isoini2  5499