Theorem rexlimivv 2432

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 2427	. 2 ⊢ (x ∈ A → (∃y ∈ B φ → ψ))
3	2	rexlimiv 2421	1 ⊢ (∃x ∈ A ∃y ∈ B φ → ψ)

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390  ∃wrex 2301

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-ral 2305  df-rex 2306

This theorem is referenced by:  opelxp  4317  f1o2ndf1  5791  xpdom2  6241  distrlem5prl  6562  distrlem5pru  6563  mulid1  6822  cnegex  6986  recexap  7416  creur  7692  creui  7693  cju  7694  elz2  8088  qre  8336  qaddcl  8346  qnegcl  8347  qmulcl  8348  qreccl  8351  replim  9087