Theorem rexlimivv 2654

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 2648	. 2 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ 𝐵 𝜑 → 𝜓))
3	2	rexlimiv 2642	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2200  ∃wrex 2509

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-ral 2513  df-rex 2514

This theorem is referenced by:  opelxp  4749  f1o2ndf1  6380  xpdom2  6998  distrlem5prl  7784  distrlem5pru  7785  mulrid  8154  cnegex  8335  recexap  8811  creur  9117  creui  9118  cju  9119  elz2  9529  qre  9832  qaddcl  9842  qnegcl  9843  qmulcl  9844  qreccl  9849  elpqb  9857  fundm2domnop0  11080  replim  11385  prodmodc  12104  odd2np1  12399  opoe  12421  omoe  12422  opeo  12423  omeo  12424  qredeu  12634  pythagtriplem1  12803  pcz  12870  4sqlem1  12926  4sqlem2  12927  4sqlem4  12930  mul4sq  12932  txuni2  14945  blssioo  15242  tgioo  15243  elply  15423  2sqlem2  15809  mul2sq  15810  2sqlem7  15815  upgredgpr  15962