Theorem rexlimivv 2620

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 2614	. 2 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝜑 → 𝜓))
3	2	rexlimiv 2608	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2167  ∃wrex 2476

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-ral 2480  df-rex 2481

This theorem is referenced by:  opelxp  4694  f1o2ndf1  6295  xpdom2  6899  distrlem5prl  7670  distrlem5pru  7671  mulrid  8040  cnegex  8221  recexap  8697  creur  9003  creui  9004  cju  9005  elz2  9414  qre  9716  qaddcl  9726  qnegcl  9727  qmulcl  9728  qreccl  9733  elpqb  9741  replim  11041  prodmodc  11760  odd2np1  12055  opoe  12077  omoe  12078  opeo  12079  omeo  12080  qredeu  12290  pythagtriplem1  12459  pcz  12526  4sqlem1  12582  4sqlem2  12583  4sqlem4  12586  mul4sq  12588  txuni2  14576  blssioo  14873  tgioo  14874  elply  15054  2sqlem2  15440  mul2sq  15441  2sqlem7  15446