Theorem rexlimdvv 2417

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 246	. . 3 ⊢ ((φ ∧ x ∈ A) → (y ∈ B → (ψ → χ)))
3	2	rexlimdv 2410	. 2 ⊢ ((φ ∧ x ∈ A) → (∃y ∈ B ψ → χ))
4	3	rexlimdva 2411	1 ⊢ (φ → (∃x ∈ A ∃y ∈ B ψ → χ))

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:  rexlimdvva  2418  f1oiso2  5391  genpcdl  6374  genpcuu  6375  distrlem1prl  6421  distrlem1pru  6422  distrlem5prl  6425  distrlem5pru  6426  recexprlemss1l  6469  recexprlemss1u  6470