Theorem rexlimi 2576

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 2519	. 2 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)
3		rexlimi.1	. . 3 ⊢ Ⅎ𝑥𝜓
4	3	r19.23 2574	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))
5	2, 4	mpbi 144	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1448   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-ral 2449  df-rex 2450

This theorem is referenced by:  rexlimiv  2577  r19.29af2  2606  triun  4093  reusv1  4436  reusv3  4438  onintrab2im  4495  fun11iun  5453  fisumcom2  11379  fprodcom2fi  11567