Theorem rexlimivv 2416

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 2411	. 2 ⊢ (x ∈ A → (∃y ∈ B φ → ψ))
3	2	rexlimiv 2405	1 ⊢ (∃x ∈ A ∃y ∈ B φ → ψ)

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1374  ∃wrex 2285

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 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-ial 1409  ax-i5r 1410

This theorem depends on definitions:  df-bi 110  df-nf 1330  df-ral 2289  df-rex 2290

This theorem is referenced by:  opelxp  4301  f1o2ndf1  5772  distrlem5prl  6425  distrlem5pru  6426  mulid1  6626  cnegex  6782