Theorem rexlimivv 2610

Description: Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 17-Feb-2004.)

Hypothesis

Ref	Expression
rexlimivv.1	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 → 𝜓))

Assertion

Ref	Expression
rexlimivv	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → 𝜓)

Distinct variable groups:   𝑥,𝑦,𝜓   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem rexlimivv

Step	Hyp	Ref	Expression
1		rexlimivv.1	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜑 → 𝜓))
2	1	rexlimdva 2604	. 2 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝜑 → 𝜓))
3	2	rexlimiv 2598	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2158  ∃wrex 2466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-4 1520  ax-17 1536  ax-ial 1544  ax-i5r 1545

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-ral 2470  df-rex 2471

This theorem is referenced by:  opelxp  4668  f1o2ndf1  6242  xpdom2  6844  distrlem5prl  7598  distrlem5pru  7599  mulrid  7967  cnegex  8148  recexap  8623  creur  8929  creui  8930  cju  8931  elz2  9337  qre  9638  qaddcl  9648  qnegcl  9649  qmulcl  9650  qreccl  9655  elpqb  9662  replim  10881  prodmodc  11599  odd2np1  11891  opoe  11913  omoe  11914  opeo  11915  omeo  11916  qredeu  12110  pythagtriplem1  12278  pcz  12344  4sqlem1  12399  4sqlem2  12400  4sqlem4  12403  mul4sq  12405  txuni2  13996  blssioo  14285  tgioo  14286  2sqlem2  14702  mul2sq  14703  2sqlem7  14708