Theorem rexlimiva 2734

Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 18-Dec-2006.)

Hypothesis

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

Assertion

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

Distinct variable group:   ψ,x

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

Proof of Theorem rexlimiva

Step	Hyp	Ref	Expression
1		rexlimiva.1	. . 3 ⊢ ((x ∈ A ∧ φ) → ψ)
2	1	ex 423	. 2 ⊢ (x ∈ A → (φ → ψ))
3	2	rexlimiv 2733	1 ⊢ (∃x ∈ A φ → ψ)

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:  lefinlteq  4464  ltlefin  4469  ncfinraise  4482  clos1basesuc  5883  pw1fin  6170  dflec3  6222