Theorem rexlimivv 2654

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 2648	. 2 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝜑 → 𝜓))
3	2	rexlimiv 2642	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2200  ∃wrex 2509

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-ral 2513  df-rex 2514

This theorem is referenced by:  opelxp  4748  f1o2ndf1  6372  xpdom2  6986  distrlem5prl  7769  distrlem5pru  7770  mulrid  8139  cnegex  8320  recexap  8796  creur  9102  creui  9103  cju  9104  elz2  9514  qre  9816  qaddcl  9826  qnegcl  9827  qmulcl  9828  qreccl  9833  elpqb  9841  fundm2domnop0  11062  replim  11365  prodmodc  12084  odd2np1  12379  opoe  12401  omoe  12402  opeo  12403  omeo  12404  qredeu  12614  pythagtriplem1  12783  pcz  12850  4sqlem1  12906  4sqlem2  12907  4sqlem4  12910  mul4sq  12912  txuni2  14924  blssioo  15221  tgioo  15222  elply  15402  2sqlem2  15788  mul2sq  15789  2sqlem7  15794  upgredgpr  15941