Theorem reximdvai 2725

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 14-Nov-2002.)

Hypothesis

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

Assertion

Ref	Expression
reximdvai	⊢ (φ → (∃x ∈ A ψ → ∃x ∈ A χ))

Distinct variable group:   φ,x

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

Proof of Theorem reximdvai

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎxφ
2		reximdvai.1	. 2 ⊢ (φ → (x ∈ A → (ψ → χ)))
3	1, 2	reximdai 2723	1 ⊢ (φ → (∃x ∈ A ψ → ∃x ∈ A χ))

Syntax hints:   → wi 4   ∈ wcel 1710  ∃wrex 2616

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  df-rex 2621

This theorem is referenced by:  reximdv  2726  reximdva  2727  reuind  3040