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  6287  xpdom2  6891  distrlem5prl  7655  distrlem5pru  7656  mulrid  8025  cnegex  8206  recexap  8682  creur  8988  creui  8989  cju  8990  elz2  9399  qre  9701  qaddcl  9711  qnegcl  9712  qmulcl  9713  qreccl  9718  elpqb  9726  replim  11026  prodmodc  11745  odd2np1  12040  opoe  12062  omoe  12063  opeo  12064  omeo  12065  qredeu  12275  pythagtriplem1  12444  pcz  12511  4sqlem1  12567  4sqlem2  12568  4sqlem4  12571  mul4sq  12573  txuni2  14502  blssioo  14799  tgioo  14800  elply  14980  2sqlem2  15366  mul2sq  15367  2sqlem7  15372