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  4693  f1o2ndf1  6286  xpdom2  6890  distrlem5prl  7653  distrlem5pru  7654  mulrid  8023  cnegex  8204  recexap  8680  creur  8986  creui  8987  cju  8988  elz2  9397  qre  9699  qaddcl  9709  qnegcl  9710  qmulcl  9711  qreccl  9716  elpqb  9724  replim  11024  prodmodc  11743  odd2np1  12038  opoe  12060  omoe  12061  opeo  12062  omeo  12063  qredeu  12265  pythagtriplem1  12434  pcz  12501  4sqlem1  12557  4sqlem2  12558  4sqlem4  12561  mul4sq  12563  txuni2  14492  blssioo  14789  tgioo  14790  elply  14970  2sqlem2  15356  mul2sq  15357  2sqlem7  15362