Theorem rexlimivv 2656

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 2650	. 2 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝜑 → 𝜓))
3	2	rexlimiv 2644	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2202  ∃wrex 2511

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-ral 2515  df-rex 2516

This theorem is referenced by:  opelxp  4755  f1o2ndf1  6393  xpdom2  7015  distrlem5prl  7806  distrlem5pru  7807  mulrid  8176  cnegex  8357  recexap  8833  creur  9139  creui  9140  cju  9141  elz2  9551  qre  9859  qaddcl  9869  qnegcl  9870  qmulcl  9871  qreccl  9876  elpqb  9884  fundm2domnop0  11113  replim  11437  prodmodc  12157  odd2np1  12452  opoe  12474  omoe  12475  opeo  12476  omeo  12477  qredeu  12687  pythagtriplem1  12856  pcz  12923  4sqlem1  12979  4sqlem2  12980  4sqlem4  12983  mul4sq  12985  txuni2  14999  blssioo  15296  tgioo  15297  elply  15477  2sqlem2  15863  mul2sq  15864  2sqlem7  15869  upgredgpr  16019