Theorem rexlimdvv 2655

Description: Inference from Theorem 19.23 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Jul-2004.)

Hypothesis

Ref	Expression
rexlimdvv.1	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒)))

Assertion

Ref	Expression
rexlimdvv	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → 𝜒))

Distinct variable groups:   𝑥,𝑦,𝜑   𝜒,𝑥,𝑦   𝑦,𝐴

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)

Proof of Theorem rexlimdvv

Step	Hyp	Ref	Expression
1		rexlimdvv.1	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝜓 → 𝜒)))
2	1	expdimp 259	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 → (𝜓 → 𝜒)))
3	2	rexlimdv 2647	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ 𝐵 𝜓 → 𝜒))
4	3	rexlimdva 2648	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓 → 𝜒))

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:  rexlimdvva  2656  f1oiso2  5957  rex2dom  6979  xpdom2  6998  genpcdl  7714  genpcuu  7715  distrlem1prl  7777  distrlem1pru  7778  distrlem5prl  7781  distrlem5pru  7782  recexprlemss1l  7830  recexprlemss1u  7831  qaddcl  9838  qmulcl  9840  summodc  11902  dvdsgcd  12541  gcddiv  12548  pceu  12826  pcqcl  12837  txcnp  14953  blssps  15109  blss  15110  tgqioo  15237  upgredg2vtx  15954