Theorem rexlimivv 2744

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 2739	. 2 ⊢ (x ∈ A → (∃y ∈ B φ → ψ))
3	2	rexlimiv 2733	1 ⊢ (∃x ∈ A ∃y ∈ B φ → ψ)

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:  2reu5  3045  tfin11  4494  peano4nc  6151  sbth  6207  nclenc  6223  lenc  6224  letc  6232  ce2le  6234