Theorem rexlimdvaa 2740

Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Mario Carneiro, 15-Jun-2016.)

Hypothesis

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

Assertion

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

Distinct variable groups:   φ,x   χ,x

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

Proof of Theorem rexlimdvaa

Step	Hyp	Ref	Expression
1		rexlimdvaa.1	. . 3 ⊢ ((φ ∧ (x ∈ A ∧ ψ)) → χ)
2	1	expr 598	. 2 ⊢ ((φ ∧ x ∈ A) → (ψ → χ))
3	2	rexlimdva 2739	1 ⊢ (φ → (∃x ∈ A ψ → χ))

Syntax hints:   → wi 4   ∧ wa 358   ∈ 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-6 1729  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:  rexlimddv  2743