Theorem rexlimi 2732

Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 30-Nov-2003.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Hypotheses

Ref	Expression
rexlimi.1	⊢ Ⅎxψ
rexlimi.2	⊢ (x ∈ A → (φ → ψ))

Assertion

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

Proof of Theorem rexlimi

Step	Hyp	Ref	Expression
1		rexlimi.2	. . 3 ⊢ (x ∈ A → (φ → ψ))
2	1	rgen 2680	. 2 ⊢ ∀x ∈ A (φ → ψ)
3		rexlimi.1	. . 3 ⊢ Ⅎxψ
4	3	r19.23 2730	. 2 ⊢ (∀x ∈ A (φ → ψ) ↔ (∃x ∈ A φ → ψ))
5	2, 4	mpbi 199	1 ⊢ (∃x ∈ A φ → ψ)

This proof depends on syntax axioms:   → wi 4  Ⅎwnf 1544   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616

This proof depends on 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 proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2620  df-rex 2621

This theorem is used by:  rexlimiv  2733  fun11iun  5306