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  7672  distrlem5pru  7673  mulrid  8042  cnegex  8223  recexap  8699  creur  9005  creui  9006  cju  9007  elz2  9416  qre  9718  qaddcl  9728  qnegcl  9729  qmulcl  9730  qreccl  9735  elpqb  9743  replim  11043  prodmodc  11762  odd2np1  12057  opoe  12079  omoe  12080  opeo  12081  omeo  12082  qredeu  12292  pythagtriplem1  12461  pcz  12528  4sqlem1  12584  4sqlem2  12585  4sqlem4  12588  mul4sq  12590  txuni2  14600  blssioo  14897  tgioo  14898  elply  15078  2sqlem2  15464  mul2sq  15465  2sqlem7  15470