Theorem rexlimdvv 2745

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 426	. . 3 ⊢ ((φ ∧ x ∈ A) → (y ∈ B → (ψ → χ)))
3	2	rexlimdv 2738	. 2 ⊢ ((φ ∧ x ∈ A) → (∃y ∈ B ψ → χ))
4	3	rexlimdva 2739	1 ⊢ (φ → (∃x ∈ A ∃y ∈ B ψ → χ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   ∈ wcel 1710  ∃wrex 2616

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2620  df-rex 2621

This theorem is used by:  rexlimdvva  2746  ncfinraise  4482  ncfinlower  4484  nnpw1ex  4485  nnpweq  4524  sfinltfin  4536  f1oiso2  5501  addcdi  6251