Theorem rexlimi 2580

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 2523	. 2 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓)
3		rexlimi.1	. . 3 ⊢ Ⅎ𝑥𝜓
4	3	r19.23 2578	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓))
5	2, 4	mpbi 144	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4  Ⅎwnf 1453   ∈ wcel 2141  ∀wral 2448  ∃wrex 2449

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-ral 2453  df-rex 2454

This theorem is referenced by:  rexlimiv  2581  r19.29af2  2610  triun  4100  reusv1  4443  reusv3  4445  onintrab2im  4502  fun11iun  5463  fisumcom2  11401  fprodcom2fi  11589