Theorem rexlimiv 2577

Description: Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 20-Nov-1994.)

Hypothesis

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

Assertion

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

Distinct variable group:   𝜓,𝑥

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

Proof of Theorem rexlimiv

Step	Hyp	Ref	Expression
1		nfv 1516	. 2 ⊢ Ⅎ𝑥𝜓
2		rexlimiv.1	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	1, 2	rexlimi 2576	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∈ wcel 2136  ∃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-17 1514  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:  rexlimiva  2578  rexlimivw  2579  rexlimivv  2589  r19.36av  2617  r19.44av  2625  r19.45av  2626  rexn0  3507  uniss2  3820  elres  4920  ssimaex  5547  mpoexw  6181  tfrlem5  6282  tfrlem8  6286  ecoptocl  6588  mapsn  6656  elixpsn  6701  ixpsnf1o  6702  findcard  6854  findcard2  6855  findcard2s  6856  fiintim  6894  prnmaddl  7431  0re  7899  cnegexlem2  8074  0cnALT  8088  bndndx  9113  uzn0  9481  ublbneg  9551  rexanuz2  10933  opnneiid  12804  2sqlem2  13591  bj-inf2vnlem2  13853